(*  Title:      HOL/Auth/n_german_lemma_on_inv__6.thy
    Author:     Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
    Copyright    2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)

(*header{*The n_german Protocol Case Study*}*)

theory n_german_lemma_on_inv__6 imports n_german_base
begin
section{*All lemmas on causal relation between inv__6 and some rule r*}
lemma n_SendInv__part__0Vsinv__6:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__0  i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInv__part__0  i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendInv__part__1Vsinv__6:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__1  i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInv__part__1  i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendInvAckVsinv__6:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInvAck  i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInvAck  i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendGntSVsinv__6:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntS  i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntS  i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendGntEVsinv__6:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntE N i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntE N i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P3 s"
  apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Para (Ident ''ShrSet'') p__Inv2)) (Const false)) (neg (eqn (IVar (Field (Para (Ident ''Cache'') p__Inv2) ''State'')) (Const I)))))" in exI, auto) done
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_RecvGntSVsinv__6:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntS  i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvGntS  i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P3 s"
  apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Para (Ident ''Chan2'') p__Inv0) ''Cmd'')) (Const GntE)) (eqn (IVar (Field (Para (Ident ''Chan2'') p__Inv2) ''Cmd'')) (Const GntS))))" in exI, auto) done
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_RecvGntEVsinv__6:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntE  i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvGntE  i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P3 s"
  apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Para (Ident ''Chan2'') p__Inv2) ''Cmd'')) (Const GntE)) (eqn (IVar (Field (Para (Ident ''Chan2'') p__Inv0) ''Cmd'')) (Const GntE))))" in exI, auto) done
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendReqE__part__1Vsinv__6:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqE__part__1  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_StoreVsinv__6:
  assumes a1: "\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store  i d" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_RecvInvAckVsinv__6:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvInvAck  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_RecvReqEVsinv__6:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvReqE N i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_SendReqE__part__0Vsinv__6:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqE__part__0  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_SendReqSVsinv__6:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqS  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_RecvReqSVsinv__6:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvReqS N i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__6  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

end


(*  Title:      HOL/Auth/Recur.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge
*)

section{*The Otway-Bull Recursive Authentication Protocol*}

theory Recur imports Public begin

text{*End marker for message bundles*}
abbreviation
  END :: "msg" where
  "END == Number 0"

(*Two session keys are distributed to each agent except for the initiator,
        who receives one.
  Perhaps the two session keys could be bundled into a single message.
*)
inductive_set (*Server's response to the nested message*)
  respond :: "event list => (msg*msg*key)set"
  for evs :: "event list"
  where
   One:  "Key KAB \<notin> used evs
          ==> (Hash[Key(shrK A)] {|Agent A, Agent B, Nonce NA, END|},
               {|Crypt (shrK A) {|Key KAB, Agent B, Nonce NA|}, END|},
               KAB)   \<in> respond evs"

    (*The most recent session key is passed up to the caller*)
 | Cons: "[| (PA, RA, KAB) \<in> respond evs;
             Key KBC \<notin> used evs;  Key KBC \<notin> parts {RA};
             PA = Hash[Key(shrK A)] {|Agent A, Agent B, Nonce NA, P|} |]
          ==> (Hash[Key(shrK B)] {|Agent B, Agent C, Nonce NB, PA|},
               {|Crypt (shrK B) {|Key KBC, Agent C, Nonce NB|},
                 Crypt (shrK B) {|Key KAB, Agent A, Nonce NB|},
                 RA|},
               KBC)
              \<in> respond evs"

(*Induction over "respond" can be difficult due to the complexity of the
  subgoals.  Set "responses" captures the general form of certificates.
*)
inductive_set
  responses :: "event list => msg set"
  for evs :: "event list"
  where
    (*Server terminates lists*)
   Nil:  "END \<in> responses evs"

 | Cons: "[| RA \<in> responses evs;  Key KAB \<notin> used evs |]
          ==> {|Crypt (shrK B) {|Key KAB, Agent A, Nonce NB|},
                RA|}  \<in> responses evs"

inductive_set recur :: "event list set"
  where
         (*Initial trace is empty*)
   Nil:  "[] \<in> recur"

         (*The spy MAY say anything he CAN say.  Common to
           all similar protocols.*)
 | Fake: "[| evsf \<in> recur;  X \<in> synth (analz (knows Spy evsf)) |]
          ==> Says Spy B X  # evsf \<in> recur"

         (*Alice initiates a protocol run.
           END is a placeholder to terminate the nesting.*)
 | RA1:  "[| evs1 \<in> recur;  Nonce NA \<notin> used evs1 |]
          ==> Says A B (Hash[Key(shrK A)] {|Agent A, Agent B, Nonce NA, END|})
              # evs1 \<in> recur"

         (*Bob's response to Alice's message.  C might be the Server.
           We omit PA = {|XA, Agent A, Agent B, Nonce NA, P|} because
           it complicates proofs, so B may respond to any message at all!*)
 | RA2:  "[| evs2 \<in> recur;  Nonce NB \<notin> used evs2;
             Says A' B PA \<in> set evs2 |]
          ==> Says B C (Hash[Key(shrK B)] {|Agent B, Agent C, Nonce NB, PA|})
              # evs2 \<in> recur"

         (*The Server receives Bob's message and prepares a response.*)
 | RA3:  "[| evs3 \<in> recur;  Says B' Server PB \<in> set evs3;
             (PB,RB,K) \<in> respond evs3 |]
          ==> Says Server B RB # evs3 \<in> recur"

         (*Bob receives the returned message and compares the Nonces with
           those in the message he previously sent the Server.*)
 | RA4:  "[| evs4 \<in> recur;
             Says B  C {|XH, Agent B, Agent C, Nonce NB,
                         XA, Agent A, Agent B, Nonce NA, P|} \<in> set evs4;
             Says C' B {|Crypt (shrK B) {|Key KBC, Agent C, Nonce NB|},
                         Crypt (shrK B) {|Key KAB, Agent A, Nonce NB|},
                         RA|} \<in> set evs4 |]
          ==> Says B A RA # evs4 \<in> recur"

   (*No "oops" message can easily be expressed.  Each session key is
     associated--in two separate messages--with two nonces.  This is
     one try, but it isn't that useful.  Re domino attack, note that
     Recur.thy proves that each session key is secure provided the two
     peers are, even if there are compromised agents elsewhere in
     the chain.  Oops cases proved using parts_cut, Key_in_keysFor_parts,
     etc.

   Oops:  "[| evso \<in> recur;  Says Server B RB \<in> set evso;
              RB \<in> responses evs';  Key K \<in> parts {RB} |]
           ==> Notes Spy {|Key K, RB|} # evso \<in> recur"
  *)

declare Says_imp_knows_Spy [THEN analz.Inj, dest]
declare parts.Body  [dest]
declare analz_into_parts [dest]
declare Fake_parts_insert_in_Un  [dest]

(** Possibility properties: traces that reach the end
        ONE theorem would be more elegant and faster!
        By induction on a list of agents (no repetitions)
**)

text{*Simplest case: Alice goes directly to the server*}
lemma "Key K \<notin> used []
       ==> \<exists>NA. \<exists>evs \<in> recur.
              Says Server A {|Crypt (shrK A) {|Key K, Agent Server, Nonce NA|},
                    END|}  \<in> set evs"
apply (intro exI bexI)
apply (rule_tac [2] recur.Nil [THEN recur.RA1,
                             THEN recur.RA3 [OF _ _ respond.One]])
apply (possibility, simp add: used_Cons)
done

text{*Case two: Alice, Bob and the server*}
lemma "[| Key K \<notin> used []; Key K' \<notin> used []; K \<noteq> K';
          Nonce NA \<notin> used []; Nonce NB \<notin> used []; NA < NB |]
       ==> \<exists>NA. \<exists>evs \<in> recur.
        Says B A {|Crypt (shrK A) {|Key K, Agent B, Nonce NA|},
                   END|}  \<in> set evs"
apply (intro exI bexI)
apply (rule_tac [2]
          recur.Nil
           [THEN recur.RA1 [of _ NA],
            THEN recur.RA2 [of _ NB],
            THEN recur.RA3 [OF _ _ respond.One
                                     [THEN respond.Cons [of _ _ K _ K']]],
            THEN recur.RA4], possibility)
apply (auto simp add: used_Cons)
done

(*Case three: Alice, Bob, Charlie and the server Rather slow (5 seconds)*)
lemma "[| Key K \<notin> used []; Key K' \<notin> used [];
          Key K'' \<notin> used []; K \<noteq> K'; K' \<noteq> K''; K \<noteq> K'';
          Nonce NA \<notin> used []; Nonce NB \<notin> used []; Nonce NC \<notin> used [];
          NA < NB; NB < NC |]
       ==> \<exists>K. \<exists>NA. \<exists>evs \<in> recur.
             Says B A {|Crypt (shrK A) {|Key K, Agent B, Nonce NA|},
                        END|}  \<in> set evs"
apply (intro exI bexI)
apply (rule_tac [2]
          recur.Nil [THEN recur.RA1,
                     THEN recur.RA2, THEN recur.RA2,
                     THEN recur.RA3
                          [OF _ _ respond.One
                                  [THEN respond.Cons, THEN respond.Cons]],
                     THEN recur.RA4, THEN recur.RA4])
apply basic_possibility
apply (tactic "DEPTH_SOLVE (swap_res_tac @{context} [refl, conjI, disjCI] 1)")
done

lemma respond_imp_not_used: "(PA,RB,KAB) \<in> respond evs ==> Key KAB \<notin> used evs"
by (erule respond.induct, simp_all)

lemma Key_in_parts_respond [rule_format]:
   "[| Key K \<in> parts {RB};  (PB,RB,K') \<in> respond evs |] ==> Key K \<notin> used evs"
apply (erule rev_mp, erule respond.induct)
apply (auto dest: Key_not_used respond_imp_not_used)
done

text{*Simple inductive reasoning about responses*}
lemma respond_imp_responses:
     "(PA,RB,KAB) \<in> respond evs ==> RB \<in> responses evs"
apply (erule respond.induct)
apply (blast intro!: respond_imp_not_used responses.intros)+
done

(** For reasoning about the encrypted portion of messages **)

lemmas RA2_analz_spies = Says_imp_spies [THEN analz.Inj]

lemma RA4_analz_spies:
     "Says C' B {|Crypt K X, X', RA|} \<in> set evs ==> RA \<in> analz (spies evs)"
by blast

(*RA2_analz... and RA4_analz... let us treat those cases using the same
  argument as for the Fake case.  This is possible for most, but not all,
  proofs: Fake does not invent new nonces (as in RA2), and of course Fake
  messages originate from the Spy. *)

lemmas RA2_parts_spies =  RA2_analz_spies [THEN analz_into_parts]
lemmas RA4_parts_spies =  RA4_analz_spies [THEN analz_into_parts]

(** Theorems of the form X \<notin> parts (spies evs) imply that NOBODY
    sends messages containing X! **)

(** Spy never sees another agent's shared key! (unless it's bad at start) **)

lemma Spy_see_shrK [simp]:
     "evs \<in> recur ==> (Key (shrK A) \<in> parts (spies evs)) = (A \<in> bad)"
apply (erule recur.induct, auto)
txt{*RA3.  It's ugly to call auto twice, but it seems necessary.*}
apply (auto dest: Key_in_parts_respond simp add: parts_insert_spies)
done

lemma Spy_analz_shrK [simp]:
     "evs \<in> recur ==> (Key (shrK A) \<in> analz (spies evs)) = (A \<in> bad)"
by auto

lemma Spy_see_shrK_D [dest!]:
     "[|Key (shrK A) \<in> parts (knows Spy evs);  evs \<in> recur|] ==> A \<in> bad"
by (blast dest: Spy_see_shrK)

(*** Proofs involving analz ***)

(** Session keys are not used to encrypt other session keys **)

(*Version for "responses" relation.  Handles case RA3 in the theorem below.
  Note that it holds for *any* set H (not just "spies evs")
  satisfying the inductive hypothesis.*)
lemma resp_analz_image_freshK_lemma:
     "[| RB \<in> responses evs;
         \<forall>K KK. KK \<subseteq> - (range shrK) -->
                   (Key K \<in> analz (Key`KK Un H)) =
                   (K \<in> KK | Key K \<in> analz H) |]
     ==> \<forall>K KK. KK \<subseteq> - (range shrK) -->
                   (Key K \<in> analz (insert RB (Key`KK Un H))) =
                   (K \<in> KK | Key K \<in> analz (insert RB H))"
apply (erule responses.induct)
apply (simp_all del: image_insert
                add: analz_image_freshK_simps, auto)
done

text{*Version for the protocol.  Proof is easy, thanks to the lemma.*}
lemma raw_analz_image_freshK:
 "evs \<in> recur ==>
   \<forall>K KK. KK \<subseteq> - (range shrK) -->
          (Key K \<in> analz (Key`KK Un (spies evs))) =
          (K \<in> KK | Key K \<in> analz (spies evs))"
apply (erule recur.induct)
apply (drule_tac [4] RA2_analz_spies,
       drule_tac [5] respond_imp_responses,
       drule_tac [6] RA4_analz_spies, analz_freshK, spy_analz)
txt{*RA3*}
apply (simp_all add: resp_analz_image_freshK_lemma)
done

(*Instance of the lemma with H replaced by (spies evs):
   [| RB \<in> responses evs;  evs \<in> recur; |]
   ==> KK \<subseteq> - (range shrK) -->
       Key K \<in> analz (insert RB (Key`KK Un spies evs)) =
       (K \<in> KK | Key K \<in> analz (insert RB (spies evs)))
*)
lemmas resp_analz_image_freshK =
       resp_analz_image_freshK_lemma [OF _ raw_analz_image_freshK]

lemma analz_insert_freshK:
     "[| evs \<in> recur;  KAB \<notin> range shrK |]
      ==> (Key K \<in> analz (insert (Key KAB) (spies evs))) =
          (K = KAB | Key K \<in> analz (spies evs))"
by (simp del: image_insert
         add: analz_image_freshK_simps raw_analz_image_freshK)

text{*Everything that's hashed is already in past traffic. *}
lemma Hash_imp_body:
     "[| Hash {|Key(shrK A), X|} \<in> parts (spies evs);
         evs \<in> recur;  A \<notin> bad |] ==> X \<in> parts (spies evs)"
apply (erule rev_mp)
apply (erule recur.induct,
       drule_tac [6] RA4_parts_spies,
       drule_tac [5] respond_imp_responses,
       drule_tac [4] RA2_parts_spies)
txt{*RA3 requires a further induction*}
apply (erule_tac [5] responses.induct, simp_all)
txt{*Fake*}
apply (blast intro: parts_insertI)
done

(** The Nonce NA uniquely identifies A's message.
    This theorem applies to steps RA1 and RA2!

  Unicity is not used in other proofs but is desirable in its own right.
**)

lemma unique_NA:
  "[| Hash {|Key(shrK A), Agent A, B, NA, P|} \<in> parts (spies evs);
      Hash {|Key(shrK A), Agent A, B',NA, P'|} \<in> parts (spies evs);
      evs \<in> recur;  A \<notin> bad |]
    ==> B=B' & P=P'"
apply (erule rev_mp, erule rev_mp)
apply (erule recur.induct,
       drule_tac [5] respond_imp_responses)
apply (force, simp_all)
txt{*Fake*}
apply blast
apply (erule_tac [3] responses.induct)
txt{*RA1,2: creation of new Nonce*}
apply simp_all
apply (blast dest!: Hash_imp_body)+
done

(*** Lemmas concerning the Server's response
      (relations "respond" and "responses")
***)

lemma shrK_in_analz_respond [simp]:
     "[| RB \<in> responses evs;  evs \<in> recur |]
  ==> (Key (shrK B) \<in> analz (insert RB (spies evs))) = (B:bad)"
apply (erule responses.induct)
apply (simp_all del: image_insert
                add: analz_image_freshK_simps resp_analz_image_freshK, auto)
done

lemma resp_analz_insert_lemma:
     "[| Key K \<in> analz (insert RB H);
         \<forall>K KK. KK \<subseteq> - (range shrK) -->
                   (Key K \<in> analz (Key`KK Un H)) =
                   (K \<in> KK | Key K \<in> analz H);
         RB \<in> responses evs |]
     ==> (Key K \<in> parts{RB} | Key K \<in> analz H)"
apply (erule rev_mp, erule responses.induct)
apply (simp_all del: image_insert
             add: analz_image_freshK_simps resp_analz_image_freshK_lemma)
txt{*Simplification using two distinct treatments of "image"*}
apply (simp add: parts_insert2, blast)
done

lemmas resp_analz_insert =
       resp_analz_insert_lemma [OF _ raw_analz_image_freshK]

text{*The last key returned by respond indeed appears in a certificate*}
lemma respond_certificate:
     "(Hash[Key(shrK A)] {|Agent A, B, NA, P|}, RA, K) \<in> respond evs
      ==> Crypt (shrK A) {|Key K, B, NA|} \<in> parts {RA}"
apply (ind_cases "(Hash[Key (shrK A)] \<lbrace>Agent A, B, NA, P\<rbrace>, RA, K) \<in> respond evs")
apply simp_all
done

(*This unicity proof differs from all the others in the HOL/Auth directory.
  The conclusion isn't quite unicity but duplicity, in that there are two
  possibilities.  Also, the presence of two different matching messages in
  the inductive step complicates the case analysis.  Unusually for such proofs,
  the quantifiers appear to be necessary.*)
lemma unique_lemma [rule_format]:
     "(PB,RB,KXY) \<in> respond evs ==>
      \<forall>A B N. Crypt (shrK A) {|Key K, Agent B, N|} \<in> parts {RB} -->
      (\<forall>A' B' N'. Crypt (shrK A') {|Key K, Agent B', N'|} \<in> parts {RB} -->
      (A'=A & B'=B) | (A'=B & B'=A))"
apply (erule respond.induct)
apply (simp_all add: all_conj_distrib)
apply (blast dest: respond_certificate)
done

lemma unique_session_keys:
     "[| Crypt (shrK A) {|Key K, Agent B, N|} \<in> parts {RB};
         Crypt (shrK A') {|Key K, Agent B', N'|} \<in> parts {RB};
         (PB,RB,KXY) \<in> respond evs |]
      ==> (A'=A & B'=B) | (A'=B & B'=A)"
by (rule unique_lemma, auto)

(** Crucial secrecy property: Spy does not see the keys sent in msg RA3
    Does not in itself guarantee security: an attack could violate
    the premises, e.g. by having A=Spy **)

lemma respond_Spy_not_see_session_key [rule_format]:
     "[| (PB,RB,KAB) \<in> respond evs;  evs \<in> recur |]
      ==> \<forall>A A' N. A \<notin> bad & A' \<notin> bad -->
          Crypt (shrK A) {|Key K, Agent A', N|} \<in> parts{RB} -->
          Key K \<notin> analz (insert RB (spies evs))"
apply (erule respond.induct)
apply (frule_tac [2] respond_imp_responses)
apply (frule_tac [2] respond_imp_not_used)
apply (simp_all del: image_insert
                add: analz_image_freshK_simps split_ifs shrK_in_analz_respond
                     resp_analz_image_freshK parts_insert2)
txt{*Base case of respond*}
apply blast
txt{*Inductive step of respond*}
apply (intro allI conjI impI, simp_all)
txt{*by unicity, either @{term "B=Aa"} or @{term "B=A'"}, a contradiction
     if @{term "B \<in> bad"}*}
apply (blast dest: unique_session_keys respond_certificate)
apply (blast dest!: respond_certificate)
apply (blast dest!: resp_analz_insert)
done

lemma Spy_not_see_session_key:
     "[| Crypt (shrK A) {|Key K, Agent A', N|} \<in> parts (spies evs);
         A \<notin> bad;  A' \<notin> bad;  evs \<in> recur |]
      ==> Key K \<notin> analz (spies evs)"
apply (erule rev_mp)
apply (erule recur.induct)
apply (drule_tac [4] RA2_analz_spies,
       frule_tac [5] respond_imp_responses,
       drule_tac [6] RA4_analz_spies,
       simp_all add: split_ifs analz_insert_eq analz_insert_freshK)
txt{*Fake*}
apply spy_analz
txt{*RA2*}
apply blast
txt{*RA3*}
apply (simp add: parts_insert_spies)
apply (metis Key_in_parts_respond parts.Body parts.Fst resp_analz_insert
             respond_Spy_not_see_session_key usedI)
txt{*RA4*}
apply blast
done

(**** Authenticity properties for Agents ****)

text{*The response never contains Hashes*}
lemma Hash_in_parts_respond:
     "[| Hash {|Key (shrK B), M|} \<in> parts (insert RB H);
         (PB,RB,K) \<in> respond evs |]
      ==> Hash {|Key (shrK B), M|} \<in> parts H"
apply (erule rev_mp)
apply (erule respond_imp_responses [THEN responses.induct], auto)
done

text{*Only RA1 or RA2 can have caused such a part of a message to appear.
  This result is of no use to B, who cannot verify the Hash.  Moreover,
  it can say nothing about how recent A's message is.  It might later be
  used to prove B's presence to A at the run's conclusion.*}
lemma Hash_auth_sender [rule_format]:
     "[| Hash {|Key(shrK A), Agent A, Agent B, NA, P|} \<in> parts(spies evs);
         A \<notin> bad;  evs \<in> recur |]
      ==> Says A B (Hash[Key(shrK A)] {|Agent A, Agent B, NA, P|}) \<in> set evs"
apply (unfold HPair_def)
apply (erule rev_mp)
apply (erule recur.induct,
       drule_tac [6] RA4_parts_spies,
       drule_tac [4] RA2_parts_spies,
       simp_all)
txt{*Fake, RA3*}
apply (blast dest: Hash_in_parts_respond)+
done

(** These two results subsume (for all agents) the guarantees proved
    separately for A and B in the Otway-Rees protocol.
**)

text{*Certificates can only originate with the Server.*}
lemma Cert_imp_Server_msg:
     "[| Crypt (shrK A) Y \<in> parts (spies evs);
         A \<notin> bad;  evs \<in> recur |]
      ==> \<exists>C RC. Says Server C RC \<in> set evs  &
                   Crypt (shrK A) Y \<in> parts {RC}"
apply (erule rev_mp, erule recur.induct, simp_all)
txt{*Fake*}
apply blast
txt{*RA1*}
apply blast
txt{*RA2: it cannot be a new Nonce, contradiction.*}
apply blast
txt{*RA3.  Pity that the proof is so brittle: this step requires the rewriting,
       which however would break all other steps.*}
apply (simp add: parts_insert_spies, blast)
txt{*RA4*}
apply blast
done

end


theory Extra2
  imports ExtraInv VCTheoryLemmas
begin

theorem extra2: "VC2 extraInv s0 requestButton_value"
  apply(simp only: VC2_def extraInv_def)
proof
  print_state
  assume VC: " (toEnvP s0 \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl = minimalRed \<longrightarrow>
           0 < ltimeEnv s1 Ctrl \<and> ltimeEnv s1 Ctrl \<le> MINIMAL_RED_TIME_LIMIT) \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl = minimalRed \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 toEnvNum s2 s1 = ltimeEnv s1 Ctrl \<and> getPstate s2 Ctrl = green \<and> ltimeEnv s2 Ctrl = GREEN_TIME_LIMIT) \<or>
           (\<forall>s2. toEnvP s2 \<and> substate s2 s1 \<longrightarrow> getPstate s2 Ctrl = minimalRed)) \<and>
     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and> substate s2 s0 \<and> getPstate s2 Ctrl = minimalRed \<and> toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<longrightarrow>
         getPstate s1 Ctrl = minimalRed) \<and>
     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and> (\<forall>s3. toEnvP s3 \<and> substate s1 s3 \<and> substate s3 s2 \<longrightarrow> getPstate s3 Ctrl = minimalRed) \<longrightarrow>
         toEnvNum s1 s2 = ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl) \<and>
     (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and> getPstate s1 Ctrl = redAfterMinimalRed \<and> getVarBool s1 redAfterMinimalRed = NOT_PRESSED \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 getPstate s2 Ctrl = minimalRed \<and>
                 ltimeEnv s2 Ctrl = MINIMAL_RED_TIME_LIMIT \<and>
                 getVarBool s2 redAfterMinimalRed = NOT_PRESSED \<and>
                 (\<forall>s3. toEnvP s3 \<and> substate s2 s3 \<and> substate s3 s1 \<and> s2 \<noteq> s3 \<longrightarrow>
                       getPstate s3 Ctrl = redAfterMinimalRed \<and>
                       getVarBool s3 redAfterMinimalRed = NOT_PRESSED \<and> getVarBool s3 Ctrl = NOT_PRESSED))) \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl = redToGreen \<longrightarrow>
           0 < ltimeEnv s1 Ctrl \<and> ltimeEnv s1 Ctrl \<le> RED_TO_GREEN_TIME_LIMIT) \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl = redToGreen \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 toEnvNum s2 s1 = ltimeEnv s1 Ctrl \<and>
                 getPstate s2 Ctrl = redAfterMinimalRed \<and>
                 (getVarBool s2 redAfterMinimalRed = PRESSED \<or> getVarBool s2 Ctrl = PRESSED))) \<and>
     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and> substate s2 s0 \<and> toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<and> getPstate s2 Ctrl = redToGreen \<longrightarrow>
         getPstate s1 Ctrl = redToGreen) \<and>
     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and> (\<forall>s3. toEnvP s3 \<and> substate s1 s3 \<and> substate s3 s2 \<longrightarrow> getPstate s3 Ctrl = redToGreen) \<longrightarrow>
         toEnvNum s1 s2 = ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl) \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl = green \<longrightarrow>
           0 < ltimeEnv s1 Ctrl \<and> ltimeEnv s1 Ctrl \<le> GREEN_TIME_LIMIT) \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl = green \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 toEnvNum s2 s1 = ltimeEnv s2 Ctrl \<and>
                 getPstate s2 Ctrl = redToGreen \<and> ltimeEnv s2 Ctrl = RED_TO_GREEN_TIME_LIMIT)) \<and>
     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and> substate s2 s0 \<and> getPstate s2 Ctrl = green \<and> toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<longrightarrow>
         getPstate s1 Ctrl = green) \<and>
     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and> (\<forall>s3. toEnvP s3 \<and> substate s1 s3 \<and> substate s3 s2 \<longrightarrow> getPstate s3 Ctrl = green) \<longrightarrow>
         toEnvNum s1 s2 = ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl) \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl = green \<longrightarrow> getVarBool s1 minimalRed = PRESSED) \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl \<noteq> green \<longrightarrow> getVarBool s1 minimalRed = NOT_PRESSED) \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<longrightarrow>
           getPstate s1 Ctrl = minimalRed \<or>
           getPstate s1 Ctrl = redAfterMinimalRed \<or> getPstate s1 Ctrl = redToGreen \<or> getPstate s1 Ctrl = green) \<and>
     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and>
         toEnvNum s1 s2 < ltimeEnv s2 Ctrl - Ctrl \<and>
         getPstate s2 Ctrl = minimalRed \<and> getVarBool s2 redAfterMinimalRed = NOT_PRESSED \<longrightarrow>
         getVarBool s1 Ctrl = NOT_PRESSED) \<and>
     (\<forall>s2. toEnvP s2 \<and> substate s2 s0 \<and> getPstate s2 Ctrl = minimalRed \<and> getVarBool s2 redAfterMinimalRed = PRESSED \<longrightarrow>
           (\<exists>s1. toEnvP s1 \<and>
                 substate s1 s2 \<and> toEnvNum s1 s2 < ltimeEnv s2 Ctrl - Ctrl \<and> getVarBool s1 Ctrl = PRESSED)) \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl = redAfterMinimalRed \<and> getVarBool s1 redAfterMinimalRed \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 toEnvNum s2 s1 = Ctrl \<and>
                 getPstate s2 Ctrl = minimalRed \<and>
                 ltimeEnv s2 Ctrl = MINIMAL_RED_TIME_LIMIT \<and>
                 (getVarBool s2 redAfterMinimalRed \<or> getVarBool s1 Ctrl = PRESSED))) \<and>
     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl = green \<longrightarrow>
           getVarBool s1 redAfterMinimalRed = NOT_PRESSED)) \<and>
    env (setVarAny s0 requestButton_value) requestButton_value \<and>
    getPstate (setVarAny s0 requestButton_value) Ctrl = minimalRed \<and>
    getVarBool (setVarAny s0 requestButton_value) Ctrl \<and>
    MINIMAL_RED_TIME_LIMIT
    \<le> ltimeEnv (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl "
  then obtain ei0: "toEnvP s0" and
 ei1: " (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and>
           getPstate s1 Ctrl = minimalRed \<longrightarrow>
           0 < ltimeEnv s1 Ctrl \<and>
           ltimeEnv s1 Ctrl
           \<le> MINIMAL_RED_TIME_LIMIT)"
 and ei2: "     (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and>
           getPstate s1 Ctrl = minimalRed \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 toEnvNum s2 s1 = ltimeEnv s1 Ctrl \<and>
                 getPstate s2 Ctrl = green \<and>
                 ltimeEnv s2 Ctrl =
                 GREEN_TIME_LIMIT) \<or>
           (\<forall>s2. toEnvP s2 \<and> substate s2 s1 \<longrightarrow>
                 getPstate s2 Ctrl = minimalRed))"
and ei3:"    (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and>
         getPstate s2 Ctrl = minimalRed \<and>
         toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<longrightarrow>
         getPstate s1 Ctrl = minimalRed)"
 and ei4: "     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and>
         (\<forall>s3. toEnvP s3 \<and>
               substate s1 s3 \<and> substate s3 s2 \<longrightarrow>
               getPstate s3 Ctrl = minimalRed) \<longrightarrow>
         toEnvNum s1 s2 =
         ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl)"
 and ei5: " (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and> getPstate s1 Ctrl = redAfterMinimalRed \<and> getVarBool s1 redAfterMinimalRed = NOT_PRESSED \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 getPstate s2 Ctrl = minimalRed \<and>
                 ltimeEnv s2 Ctrl = MINIMAL_RED_TIME_LIMIT \<and>
                 getVarBool s2 redAfterMinimalRed = NOT_PRESSED \<and>
                 (\<forall>s3. toEnvP s3 \<and> substate s2 s3 \<and> substate s3 s1 \<and> s2 \<noteq> s3 \<longrightarrow>
                       getPstate s3 Ctrl = redAfterMinimalRed \<and>
                       getVarBool s3 redAfterMinimalRed = NOT_PRESSED \<and> getVarBool s3 Ctrl = NOT_PRESSED)))" and ei6:
"     (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and>
           getPstate s1 Ctrl = redToGreen \<longrightarrow>
           0 < ltimeEnv s1 Ctrl \<and>
           ltimeEnv s1 Ctrl
           \<le> RED_TO_GREEN_TIME_LIMIT)"
and ei7: "     (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and>
           getPstate s1 Ctrl = redToGreen \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 toEnvNum s2 s1 = ltimeEnv s1 Ctrl \<and>
                 getPstate s2 Ctrl =
                 redAfterMinimalRed \<and>
                 (getVarBool s2 redAfterMinimalRed =
                  PRESSED \<or>
                  getVarBool s2 Ctrl = PRESSED)))"
and ei8: "     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and>
         toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<and>
         getPstate s2 Ctrl = redToGreen \<longrightarrow>
         getPstate s1 Ctrl = redToGreen)"
    and ei9: "     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and>
         (\<forall>s3. toEnvP s3 \<and>
               substate s1 s3 \<and> substate s3 s2 \<longrightarrow>
               getPstate s3 Ctrl = redToGreen) \<longrightarrow>
         toEnvNum s1 s2 =
         ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl)"
 and ei10:
"     (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and>
           getPstate s1 Ctrl = green \<longrightarrow>
           0 < ltimeEnv s1 Ctrl \<and>
           ltimeEnv s1 Ctrl \<le> GREEN_TIME_LIMIT)"
     and ei11: "     (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and>
           getPstate s1 Ctrl = green \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 toEnvNum s2 s1 = ltimeEnv s2 Ctrl \<and>
                 getPstate s2 Ctrl = redToGreen \<and>
                 ltimeEnv s2 Ctrl =
                 RED_TO_GREEN_TIME_LIMIT))"
  and ei12: "     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and>
         getPstate s2 Ctrl = green \<and>
         toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<longrightarrow>
         getPstate s1 Ctrl = green)" and ei13:
"     (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and>
         (\<forall>s3. toEnvP s3 \<and>
               substate s1 s3 \<and> substate s3 s2 \<longrightarrow>
               getPstate s3 Ctrl = green) \<longrightarrow>
         toEnvNum s1 s2 =
         ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl)"
and ei14: "     (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and>
           getPstate s1 Ctrl = green \<longrightarrow>
           getVarBool s1 minimalRed = PRESSED)" and
ei15: "     (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and>
           getPstate s1 Ctrl \<noteq> green \<longrightarrow>
           getVarBool s1 minimalRed = NOT_PRESSED)"
and ei16: "     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<longrightarrow>
           getPstate s1 Ctrl = minimalRed \<or>
           getPstate s1 Ctrl = redAfterMinimalRed \<or>
           getPstate s1 Ctrl = redToGreen \<or>
           getPstate s1 Ctrl = green)" and
ei17: " (\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2 s0 \<and>
         toEnvNum s1 s2 < ltimeEnv s2 Ctrl - Ctrl \<and>
         getPstate s2 Ctrl = minimalRed \<and> getVarBool s2 redAfterMinimalRed = NOT_PRESSED \<longrightarrow>
         getVarBool s1 Ctrl = NOT_PRESSED)" and
ei18: "     (\<forall>s2. toEnvP s2 \<and> substate s2 s0 \<and> getPstate s2 Ctrl = minimalRed \<and> getVarBool s2 redAfterMinimalRed = PRESSED \<longrightarrow>
           (\<exists>s1. toEnvP s1 \<and>
                 substate s1 s2 \<and> toEnvNum s1 s2 < ltimeEnv s2 Ctrl - Ctrl \<and> getVarBool s1 Ctrl = PRESSED))" and
ei19: "     (\<forall>s1. toEnvP s1 \<and> substate s1 s0 \<and> getPstate s1 Ctrl = redAfterMinimalRed \<and> getVarBool s1 redAfterMinimalRed \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 toEnvNum s2 s1 = Ctrl \<and>
                 getPstate s2 Ctrl = minimalRed \<and>
                 ltimeEnv s2 Ctrl = MINIMAL_RED_TIME_LIMIT \<and>
                 (getVarBool s2 redAfterMinimalRed \<or> getVarBool s1 Ctrl = PRESSED)))"
 and ei20:
"     (\<forall>s1. toEnvP s1 \<and>
           substate s1 s0 \<and>
           getPstate s1 Ctrl = green \<longrightarrow>
           getVarBool s1 redAfterMinimalRed =
           NOT_PRESSED)"
and vc: " env (setVarAny s0 requestButton_value)
     requestButton_value \<and>
    getPstate (setVarAny s0 requestButton_value)
     Ctrl =
    minimalRed \<and>
    getVarBool (setVarAny s0 requestButton_value)
     Ctrl \<and>
    MINIMAL_RED_TIME_LIMIT
    \<le> ltimeEnv
        (setVarBool
          (setVarAny s0 requestButton_value)
          redAfterMinimalRed PRESSED)
        Ctrl "
    by fastforce
   have "toEnvP
     (toEnv
       (setPstate
         (setVarBool
           (setVarAny s0 requestButton_value)
           redAfterMinimalRed PRESSED)
         Ctrl redAfterMinimalRed))" by auto
  moreover from ei1   have "  (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = minimalRed \<longrightarrow>
          0 < ltimeEnv s1 Ctrl \<and>
          ltimeEnv s1 Ctrl
          \<le> MINIMAL_RED_TIME_LIMIT)"
    by auto
  moreover from ei2 have " (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = minimalRed \<longrightarrow>
          (\<exists>s2. toEnvP s2 \<and>
                substate s2 s1 \<and>
                toEnvNum s2 s1 = ltimeEnv s1 Ctrl \<and>
                getPstate s2 Ctrl = green \<and>
                ltimeEnv s2 Ctrl =
                GREEN_TIME_LIMIT) \<or>
          (\<forall>s2. toEnvP s2 \<and> substate s2 s1 \<longrightarrow>
                getPstate s2 Ctrl = minimalRed))"
    by auto
  moreover from ei3 have " (\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate
             (setVarBool
               (setVarAny s0 requestButton_value)
               redAfterMinimalRed PRESSED)
             Ctrl redAfterMinimalRed)) \<and>
        getPstate s2 Ctrl = minimalRed \<and>
        toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<longrightarrow>
        getPstate s1 Ctrl = minimalRed)"
    by auto
  moreover from ei4 have " (\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate
             (setVarBool
               (setVarAny s0 requestButton_value)
               redAfterMinimalRed PRESSED)
             Ctrl redAfterMinimalRed)) \<and>
        (\<forall>s3. toEnvP s3 \<and>
              substate s1 s3 \<and> substate s3 s2 \<longrightarrow>
              getPstate s3 Ctrl = minimalRed) \<longrightarrow>
        toEnvNum s1 s2 =
        ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl)"
    by auto
  moreover  from ei5 have "  (\<forall>s1. toEnvP s1 \<and>
           substate s1 (toEnv
           (setPstate
             (setVarBool
               (setVarAny s0 requestButton_value)
               redAfterMinimalRed PRESSED)
             Ctrl redAfterMinimalRed))
 \<and> getPstate s1 Ctrl = redAfterMinimalRed \<and> getVarBool s1 redAfterMinimalRed = NOT_PRESSED \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 getPstate s2 Ctrl = minimalRed \<and>
                 ltimeEnv s2 Ctrl = MINIMAL_RED_TIME_LIMIT \<and>
                 getVarBool s2 redAfterMinimalRed = NOT_PRESSED \<and>
                 (\<forall>s3. toEnvP s3 \<and> substate s2 s3 \<and> substate s3 s1 \<and> s2 \<noteq> s3 \<longrightarrow>
                       getPstate s3 Ctrl = redAfterMinimalRed \<and>
                       getVarBool s3 redAfterMinimalRed = NOT_PRESSED \<and> getVarBool s3 Ctrl = NOT_PRESSED)))"
    by auto
   moreover from ei6  have "(\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = redToGreen \<longrightarrow>
          0 < ltimeEnv s1 Ctrl \<and>
          ltimeEnv s1 Ctrl
          \<le> RED_TO_GREEN_TIME_LIMIT)"
     by auto
   moreover from ei7 have "(\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = redToGreen \<longrightarrow>
          (\<exists>s2. toEnvP s2 \<and>
                substate s2 s1 \<and>
                toEnvNum s2 s1 = ltimeEnv s1 Ctrl \<and>
                getPstate s2 Ctrl =
                redAfterMinimalRed \<and>
                (getVarBool s2 redAfterMinimalRed =
                 PRESSED \<or>
                 getVarBool s2 Ctrl = PRESSED)))"
     by auto
   moreover from ei8 have "(\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate
             (setVarBool
               (setVarAny s0 requestButton_value)
               redAfterMinimalRed PRESSED)
             Ctrl redAfterMinimalRed)) \<and>
        toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<and>
        getPstate s2 Ctrl = redToGreen \<longrightarrow>
        getPstate s1 Ctrl = redToGreen)"
     by auto
   moreover from ei9 have "(\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate
             (setVarBool
               (setVarAny s0 requestButton_value)
               redAfterMinimalRed PRESSED)
             Ctrl redAfterMinimalRed)) \<and>
        (\<forall>s3. toEnvP s3 \<and>
              substate s1 s3 \<and> substate s3 s2 \<longrightarrow>
              getPstate s3 Ctrl = redToGreen) \<longrightarrow>
        toEnvNum s1 s2 =
        ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl)"
     by auto
   moreover from ei10 have " (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = green \<longrightarrow>
          0 < ltimeEnv s1 Ctrl \<and>
          ltimeEnv s1 Ctrl \<le> GREEN_TIME_LIMIT)"
     by auto
   moreover from ei11 have " \<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = green \<longrightarrow>
          (\<exists>s2. toEnvP s2 \<and>
                substate s2 s1 \<and>
                toEnvNum s2 s1 = ltimeEnv s2 Ctrl \<and>
                getPstate s2 Ctrl = redToGreen \<and>
                ltimeEnv s2 Ctrl =
                RED_TO_GREEN_TIME_LIMIT)"
     by auto
   moreover from ei12 have "(\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate
             (setVarBool
               (setVarAny s0 requestButton_value)
               redAfterMinimalRed PRESSED)
             Ctrl redAfterMinimalRed)) \<and>
        getPstate s2 Ctrl = green \<and>
        toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<longrightarrow>
        getPstate s1 Ctrl = green)"
     by auto
   moreover from ei13 have "(\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate
             (setVarBool
               (setVarAny s0 requestButton_value)
               redAfterMinimalRed PRESSED)
             Ctrl redAfterMinimalRed)) \<and>
        (\<forall>s3. toEnvP s3 \<and>
              substate s1 s3 \<and> substate s3 s2 \<longrightarrow>
              getPstate s3 Ctrl = green) \<longrightarrow>
        toEnvNum s1 s2 =
        ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl)"
     by auto
   moreover from ei14 have " (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = green \<longrightarrow>
          getVarBool s1 minimalRed = PRESSED)"
     by auto
   moreover from ei0 ei15 vc substate_refl have "(\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl \<noteq> green \<longrightarrow>
          getVarBool s1 minimalRed = NOT_PRESSED)" by auto
   moreover from ei16 have "(\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<longrightarrow>
          getPstate s1 Ctrl = minimalRed \<or>
          getPstate s1 Ctrl = redAfterMinimalRed \<or>
          getPstate s1 Ctrl = redToGreen \<or>
          getPstate s1 Ctrl = green)"
     by auto
   moreover from ei17 have "(\<forall>s1 s2.
         toEnvP s1 \<and>
         toEnvP s2 \<and>
         substate s1 s2 \<and>
         substate s2  (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<and>
         toEnvNum s1 s2 < ltimeEnv s2 Ctrl - Ctrl \<and>
         getPstate s2 Ctrl = minimalRed \<and> getVarBool s2 redAfterMinimalRed = NOT_PRESSED \<longrightarrow>
         getVarBool s1 Ctrl = NOT_PRESSED)"
     by auto
   moreover from ei18  have " (\<forall>s2. toEnvP s2 \<and> substate s2
 (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed))
 \<and> getPstate s2 Ctrl = minimalRed \<and> getVarBool s2 redAfterMinimalRed = PRESSED \<longrightarrow>
           (\<exists>s1. toEnvP s1 \<and>
                 substate s1 s2 \<and> toEnvNum s1 s2 < ltimeEnv s2 Ctrl - Ctrl \<and> getVarBool s1 Ctrl = PRESSED))"
     by auto
   moreover have " (\<forall>s1. toEnvP s1 \<and> substate s1  (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed))
 \<and> getPstate s1 Ctrl = redAfterMinimalRed \<and> getVarBool s1 redAfterMinimalRed \<longrightarrow>
           (\<exists>s2. toEnvP s2 \<and>
                 substate s2 s1 \<and>
                 toEnvNum s2 s1 = Ctrl \<and>
                 getPstate s2 Ctrl = minimalRed \<and>
                 ltimeEnv s2 Ctrl = MINIMAL_RED_TIME_LIMIT \<and>
                 (getVarBool s2 redAfterMinimalRed \<or> getVarBool s1 Ctrl = PRESSED)))"
   proof -
     from ei0 ei1  vc substate_refl toEnvNum_id  have 1: "( toEnvP s0 \<and>
                 substate s0 (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed)) \<and>
                 toEnvNum s0 (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed))
 = Ctrl \<and>
                 getPstate s0 Ctrl = minimalRed \<and>
                 ltimeEnv s0 Ctrl = MINIMAL_RED_TIME_LIMIT \<and>
                 (getVarBool s0 redAfterMinimalRed \<or> getVarBool (toEnv
             (setPstate
               (setVarBool
                 (setVarAny s0 requestButton_value)
                 redAfterMinimalRed PRESSED)
               Ctrl redAfterMinimalRed))
 Ctrl = PRESSED))"
       by auto
     show ?thesis
     proof
       fix s1
       show " toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = redAfterMinimalRed \<and> getVarBool s1 redAfterMinimalRed \<longrightarrow>
          (\<exists>s2. toEnvP s2 \<and>
                substate s2 s1 \<and>
                toEnvNum s2 s1 = Ctrl \<and>
                getPstate s2 Ctrl = minimalRed \<and>
                ltimeEnv s2 Ctrl = MINIMAL_RED_TIME_LIMIT \<and>
                (getVarBool s2 redAfterMinimalRed \<or> getVarBool s1 Ctrl = PRESSED))"
       proof cases
         assume 2: "s1 = (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed))"
          show ?thesis
          proof
            from 1 2  show "\<exists>s2. toEnvP s2 \<and>
         substate s2 s1 \<and>
         toEnvNum s2 s1 = Ctrl \<and>
         getPstate s2 Ctrl = minimalRed \<and>
         ltimeEnv s2 Ctrl = MINIMAL_RED_TIME_LIMIT \<and> (getVarBool s2 redAfterMinimalRed \<or> getVarBool s1 Ctrl = PRESSED)"
              by blast
          qed
        next
assume 2: "s1 \<noteq> (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed))"
  with ei19 show ?thesis by auto
qed
qed
qed
  moreover from ei20 have "(\<forall>s1. toEnvP s1 \<and>
           substate s1  (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
           getPstate s1 Ctrl = green \<longrightarrow>
           getVarBool s1 redAfterMinimalRed =
           NOT_PRESSED)"
    by auto
  ultimately  show "toEnvP
     (toEnv
       (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
         redAfterMinimalRed)) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = minimalRed \<longrightarrow>
          0 < ltimeEnv s1 Ctrl \<and> ltimeEnv s1 Ctrl \<le> MINIMAL_RED_TIME_LIMIT) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = minimalRed \<longrightarrow>
          (\<exists>s2. toEnvP s2 \<and>
                substate s2 s1 \<and>
                toEnvNum s2 s1 = ltimeEnv s1 Ctrl \<and> getPstate s2 Ctrl = green \<and> ltimeEnv s2 Ctrl = GREEN_TIME_LIMIT) \<or>
          (\<forall>s2. toEnvP s2 \<and> substate s2 s1 \<longrightarrow> getPstate s2 Ctrl = minimalRed)) \<and>
    (\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
             redAfterMinimalRed)) \<and>
        getPstate s2 Ctrl = minimalRed \<and> toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<longrightarrow>
        getPstate s1 Ctrl = minimalRed) \<and>
    (\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
             redAfterMinimalRed)) \<and>
        (\<forall>s3. toEnvP s3 \<and> substate s1 s3 \<and> substate s3 s2 \<longrightarrow> getPstate s3 Ctrl = minimalRed) \<longrightarrow>
        toEnvNum s1 s2 = ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = redAfterMinimalRed \<and> getVarBool s1 redAfterMinimalRed = NOT_PRESSED \<longrightarrow>
          (\<exists>s2. toEnvP s2 \<and>
                substate s2 s1 \<and>
                getPstate s2 Ctrl = minimalRed \<and>
                ltimeEnv s2 Ctrl = MINIMAL_RED_TIME_LIMIT \<and>
                getVarBool s2 redAfterMinimalRed = NOT_PRESSED \<and>
                (\<forall>s3. toEnvP s3 \<and> substate s2 s3 \<and> substate s3 s1 \<and> s2 \<noteq> s3 \<longrightarrow>
                      getPstate s3 Ctrl = redAfterMinimalRed \<and>
                      getVarBool s3 redAfterMinimalRed = NOT_PRESSED \<and> getVarBool s3 Ctrl = NOT_PRESSED))) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = redToGreen \<longrightarrow>
          0 < ltimeEnv s1 Ctrl \<and> ltimeEnv s1 Ctrl \<le> RED_TO_GREEN_TIME_LIMIT) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = redToGreen \<longrightarrow>
          (\<exists>s2. toEnvP s2 \<and>
                substate s2 s1 \<and>
                toEnvNum s2 s1 = ltimeEnv s1 Ctrl \<and>
                getPstate s2 Ctrl = redAfterMinimalRed \<and>
                (getVarBool s2 redAfterMinimalRed = PRESSED \<or> getVarBool s2 Ctrl = PRESSED))) \<and>
    (\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
             redAfterMinimalRed)) \<and>
        toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<and> getPstate s2 Ctrl = redToGreen \<longrightarrow>
        getPstate s1 Ctrl = redToGreen) \<and>
    (\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
             redAfterMinimalRed)) \<and>
        (\<forall>s3. toEnvP s3 \<and> substate s1 s3 \<and> substate s3 s2 \<longrightarrow> getPstate s3 Ctrl = redToGreen) \<longrightarrow>
        toEnvNum s1 s2 = ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = green \<longrightarrow>
          0 < ltimeEnv s1 Ctrl \<and> ltimeEnv s1 Ctrl \<le> GREEN_TIME_LIMIT) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = green \<longrightarrow>
          (\<exists>s2. toEnvP s2 \<and>
                substate s2 s1 \<and>
                toEnvNum s2 s1 = ltimeEnv s2 Ctrl \<and>
                getPstate s2 Ctrl = redToGreen \<and> ltimeEnv s2 Ctrl = RED_TO_GREEN_TIME_LIMIT)) \<and>
    (\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
             redAfterMinimalRed)) \<and>
        getPstate s2 Ctrl = green \<and> toEnvNum s1 s2 < ltimeEnv s2 Ctrl \<longrightarrow>
        getPstate s1 Ctrl = green) \<and>
    (\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
             redAfterMinimalRed)) \<and>
        (\<forall>s3. toEnvP s3 \<and> substate s1 s3 \<and> substate s3 s2 \<longrightarrow> getPstate s3 Ctrl = green) \<longrightarrow>
        toEnvNum s1 s2 = ltimeEnv s2 Ctrl - ltimeEnv s1 Ctrl) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = green \<longrightarrow>
          getVarBool s1 minimalRed = PRESSED) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl \<noteq> green \<longrightarrow>
          getVarBool s1 minimalRed = NOT_PRESSED) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<longrightarrow>
          getPstate s1 Ctrl = minimalRed \<or>
          getPstate s1 Ctrl = redAfterMinimalRed \<or> getPstate s1 Ctrl = redToGreen \<or> getPstate s1 Ctrl = green) \<and>
    (\<forall>s1 s2.
        toEnvP s1 \<and>
        toEnvP s2 \<and>
        substate s1 s2 \<and>
        substate s2
         (toEnv
           (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
             redAfterMinimalRed)) \<and>
        toEnvNum s1 s2 < ltimeEnv s2 Ctrl - Ctrl \<and>
        getPstate s2 Ctrl = minimalRed \<and> getVarBool s2 redAfterMinimalRed = NOT_PRESSED \<longrightarrow>
        getVarBool s1 Ctrl = NOT_PRESSED) \<and>
    (\<forall>s2. toEnvP s2 \<and>
          substate s2
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s2 Ctrl = minimalRed \<and> getVarBool s2 redAfterMinimalRed = PRESSED \<longrightarrow>
          (\<exists>s1. toEnvP s1 \<and>
                substate s1 s2 \<and> toEnvNum s1 s2 < ltimeEnv s2 Ctrl - Ctrl \<and> getVarBool s1 Ctrl = PRESSED)) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = redAfterMinimalRed \<and> getVarBool s1 redAfterMinimalRed \<longrightarrow>
          (\<exists>s2. toEnvP s2 \<and>
                substate s2 s1 \<and>
                toEnvNum s2 s1 = Ctrl \<and>
                getPstate s2 Ctrl = minimalRed \<and>
                ltimeEnv s2 Ctrl = MINIMAL_RED_TIME_LIMIT \<and>
                (getVarBool s2 redAfterMinimalRed \<or> getVarBool s1 Ctrl = PRESSED))) \<and>
    (\<forall>s1. toEnvP s1 \<and>
          substate s1
           (toEnv
             (setPstate (setVarBool (setVarAny s0 requestButton_value) redAfterMinimalRed PRESSED) Ctrl
               redAfterMinimalRed)) \<and>
          getPstate s1 Ctrl = green \<longrightarrow>
          getVarBool s1 redAfterMinimalRed = NOT_PRESSED)" by blast
qed


(*  Title:      HOL/Auth/n_german_lemma_on_inv__52.thy
    Author:     Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
    Copyright    2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)

header{*The n_german Protocol Case Study*}

theory n_german_lemma_on_inv__52 imports n_german_base
begin
section{*All lemmas on causal relation between inv__52 and some rule r*}
lemma n_RecvInvAckVsinv__52:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvInvAck  i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvInvAck  i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
  moreover {
    assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  moreover {
    assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
  moreover {
    assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  moreover {
    assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
  moreover {
    assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  moreover {
    assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
    have "?P2 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendGntSVsinv__52:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntS  i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntS  i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendGntEVsinv__52:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntE N i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntE N i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendReqE__part__1Vsinv__52:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqE__part__1  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_StoreVsinv__52:
  assumes a1: "\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store  i d" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_RecvGntSVsinv__52:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvGntS  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_RecvReqEVsinv__52:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvReqE N i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_RecvGntEVsinv__52:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvGntE  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_SendInvAckVsinv__52:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendInvAck  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_SendInv__part__0Vsinv__52:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendInv__part__0  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_SendReqE__part__0Vsinv__52:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqE__part__0  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_SendInv__part__1Vsinv__52:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendInv__part__1  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_SendReqSVsinv__52:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqS  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_RecvReqSVsinv__52:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvReqS N i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__52  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

end


theory unique_ids_procedure_check
  imports unique_ids
begin

inductive procedure_cannot_guess_ids :: "uniqueId set \<Rightarrow> 'ls \<Rightarrow> ('ls, 'op::valueType, 'any::valueType) procedureImpl \<Rightarrow> bool"  where
  pcgi_local:  "\<lbrakk>impl ls = LocalStep ok ls'; procedure_cannot_guess_ids uids ls' impl\<rbrakk> \<Longrightarrow>  procedure_cannot_guess_ids uids ls impl"
| pcgi_beginAtomic: "\<lbrakk>impl ls = BeginAtomic ls'; procedure_cannot_guess_ids uids ls' impl\<rbrakk> \<Longrightarrow>  procedure_cannot_guess_ids uids ls impl"
| pcgi_endAtomic:"\<lbrakk>impl ls = EndAtomic ls'; procedure_cannot_guess_ids uids ls' impl\<rbrakk> \<Longrightarrow>  procedure_cannot_guess_ids uids ls impl"
| pcgi_newId:"\<lbrakk>impl ls = NewId f; \<And>uid ls'. f uid \<triangleq> ls' \<Longrightarrow> procedure_cannot_guess_ids (uids \<union> uniqueIds uid) ls' impl
    \<rbrakk> \<Longrightarrow>  procedure_cannot_guess_ids uids ls impl"
| pcgi_dbop: "\<lbrakk>impl ls = DbOperation opr f;  uniqueIds opr \<subseteq> uids;
    \<And>res. procedure_cannot_guess_ids (uids \<union> uniqueIds res) (f res) impl
    \<rbrakk> \<Longrightarrow>  procedure_cannot_guess_ids uids ls impl"
| pcgi_return: "\<lbrakk>impl ls = Return r; uniqueIds r \<subseteq> uids\<rbrakk> \<Longrightarrow> procedure_cannot_guess_ids uids ls impl"

lemma pcgi_local_case:"\<lbrakk>procedure_cannot_guess_ids uids ls impl; impl ls = LocalStep ok ls'\<rbrakk> \<Longrightarrow> procedure_cannot_guess_ids uids ls' impl"
  by (subst(asm) procedure_cannot_guess_ids.simps, auto)
lemma pcgi_beginAtomic_case:"\<lbrakk>procedure_cannot_guess_ids uids ls impl; impl ls = BeginAtomic ls'\<rbrakk> \<Longrightarrow> procedure_cannot_guess_ids uids ls' impl"
  by (subst(asm) procedure_cannot_guess_ids.simps, auto)
lemma pcgi_endAtomic_case:"\<lbrakk>procedure_cannot_guess_ids uids ls impl; impl ls = EndAtomic ls'\<rbrakk> \<Longrightarrow> procedure_cannot_guess_ids uids ls' impl"
  by (subst(asm) procedure_cannot_guess_ids.simps, auto)
lemma pcgi_newId_case:"\<lbrakk>procedure_cannot_guess_ids uids ls impl; impl ls = NewId f\<rbrakk> \<Longrightarrow> f uid \<triangleq> ls' \<Longrightarrow> procedure_cannot_guess_ids (uids \<union> uniqueIds uid) ls' impl"
  by (subst(asm) procedure_cannot_guess_ids.simps, auto)
lemma pcgi_dbop_case1:"\<lbrakk>procedure_cannot_guess_ids uids ls impl; impl ls = DbOperation opr f\<rbrakk> \<Longrightarrow> uniqueIds opr \<subseteq> uids"
  by (subst(asm) procedure_cannot_guess_ids.simps, auto)
lemma pcgi_dbop_case2:"\<lbrakk>procedure_cannot_guess_ids uids ls impl; impl ls = DbOperation opr f\<rbrakk> \<Longrightarrow> procedure_cannot_guess_ids (uids \<union> uniqueIds res) (f res) impl"
  by (subst(asm) procedure_cannot_guess_ids.simps, auto)
lemma pcgi_return_case:"\<lbrakk>procedure_cannot_guess_ids uids ls impl; impl ls = Return r\<rbrakk> \<Longrightarrow> uniqueIds r \<subseteq> uids"
  by (subst(asm) procedure_cannot_guess_ids.simps, auto)

definition procedures_cannot_guess_ids :: "('proc::valueType \<Rightarrow> ('ls \<times> ('ls, 'op::valueType, 'any::valueType) procedureImpl)) \<Rightarrow> bool" where
  "procedures_cannot_guess_ids proc =
(\<forall>p ls impl uids. proc p = (ls, impl) \<longrightarrow>  procedure_cannot_guess_ids (uids\<union>uniqueIds p) ls impl)"

lemmas show_procedures_cannot_guess_ids = procedures_cannot_guess_ids_def[THEN iffD1, rule_format]

lemma pscgi_to_iscgi:
  assumes "procedures_cannot_guess_ids (procedure progr)"
  shows "invocations_cannot_guess_ids progr"
  unfolding invocations_cannot_guess_ids_def invocation_cannot_guess_ids_def
proof (intro allI impI)

  fix i1 tr1 a S'
  assume steps: "initialState progr ~~ tr1 @ [a] \<leadsto>* S'"

  define tr1a where "tr1a = tr1@[a]"

  from steps
  have "initialState progr ~~ tr1a \<leadsto>* S'"
    using tr1a_def by blast

  value "butlast [1,2,3::nat]"

  from this
  have "trace_outputs tr1a i1 \<subseteq> trace_inputs (butlast tr1a) i1
    \<and> (\<forall>ls impl. localState S' i1 \<triangleq> ls
        \<longrightarrow>  currentProc S' i1 \<triangleq> impl
        \<longrightarrow> procedure_cannot_guess_ids (trace_inputs tr1a i1) ls impl)"
  proof (induct rule: steps_induct)
    case initial
    then show ?case
      by (auto simp add: initialState_def trace_outputs_def)

  next
    case (step S' tr a S'')

    have IH1: "trace_outputs tr i1 \<subseteq> trace_inputs (butlast tr) i1"
      using step by auto

    hence IH1': "trace_outputs tr i1 \<subseteq> trace_inputs tr i1"
      by (induct tr rule: rev_induct, auto simp add:  trace_inputs_append)

    have IH2: "procedure_cannot_guess_ids (trace_inputs tr i1) ls impl "
      if "localState S' i1 \<triangleq> ls"
        and "currentProc S' i1 \<triangleq> impl"
      for ls impl
      using step that by blast

    show ?case
    proof (cases "get_invoc a = i1")
      case False
      have [simp]: "trace_inputs (tr @ [a]) i1 = trace_inputs tr i1"
        by (simp add: False trace_inputs_def)
      have [simp]: "trace_outputs (tr @ [a]) i1 = trace_outputs tr i1"
        by (simp add: False trace_outputs_def)

      show ?thesis
        using IH1' IH2 apply (auto simp add: step)
        by (metis False IH2 prod.collapse step.step unchangedInTransaction(1) unchangedInTransaction(2))
    next
      case True

      from `S' ~~ a \<leadsto> S''`
      have "action_outputs (get_action a) \<subseteq> trace_inputs tr i1"
      proof cases
        case (dbop i ls f Op ls' t c res vis)
        have pcgi: "procedure_cannot_guess_ids (trace_inputs tr i1) ls f"
          using True local.dbop(1) local.dbop(3) local.dbop(4) step.IH by auto

        from pcgi_dbop_case1[OF pcgi `f ls = DbOperation Op ls'`]
        have " uniqueIds Op \<subseteq> trace_inputs tr i1" .
        then show ?thesis
          by (auto simp add: `a = (i, ADbOp c Op res)` trace_outputs_empty)
      next
        case (return i ls f res)
        have pcgi: "procedure_cannot_guess_ids (trace_inputs tr i1) ls f"
          using IH2 True local.return(1) local.return(3) local.return(4) by auto

        from pcgi_return_case[OF pcgi]
        show ?thesis
          by (auto simp add: return)
      qed auto

      have [simp]: "(trace_inputs (tr @ [(i, action)]) i1)
          = trace_inputs tr i1 \<union> (if i = i1 then action_inputs action else {})" for i action
        by (auto simp add: trace_inputs_append trace_inputs_cons trace_inputs_empty)

      from `S' ~~ a \<leadsto> S''`
      have "procedure_cannot_guess_ids (trace_inputs (tr @ [a]) i1) ls impl"
        if l1: "localState S'' i1 \<triangleq> ls"
        and l2: "currentProc S'' i1 \<triangleq> impl"
      for ls impl
      proof cases
        case (local i ls f ok ls')
        then show ?thesis
          using IH2 l1 l2 apply (auto simp add:  split: if_splits)
          by (meson pcgi_local_case)
      next
        case (newId i ls f ls' uid uidv ls'')
        then show ?thesis
          using IH2 l1 l2 apply (auto simp add:  split: if_splits)
          using pcgi_newId_case by fastforce
      next
        case (beginAtomic i ls f ls' t vis snapshot)
        then show ?thesis
          using IH2 l1 l2 apply (auto simp add:  split: if_splits)
          by (meson pcgi_beginAtomic_case)
      next
        case (endAtomic i ls f ls' t)
        then show ?thesis
          using IH2 l1 l2 apply (auto simp add:  split: if_splits)
          by (meson pcgi_endAtomic_case)
      next
        case (dbop i ls f Op ls' t c res vis)
        then show ?thesis
          using IH2 l1 l2 apply (auto simp add:  split: if_splits)
          by (meson pcgi_dbop_case2)
      next
        case (invocation i proc initialState impl)
        from  `procedures_cannot_guess_ids (procedure progr)`
        have "procedure_cannot_guess_ids (uniqueIds proc) initialState impl"
          apply (auto simp add: procedures_cannot_guess_ids_def)
          by (metis (no_types, lifting) state.select_convs(1) initialState_def local.invocation(4) step.steps steps_do_not_change_prog sup.idem)

        from invocation
        show ?thesis
          using IH2 l1 l2 apply (auto simp add:  split: if_splits)
          by (metis (no_types, lifting) assms state.select_convs(1) initialState_def show_procedures_cannot_guess_ids step.steps steps_do_not_change_prog)
      next
        case (return i ls f res)
        then show ?thesis
          using IH2 l1 l2 by (auto simp add:  split: if_splits)
      next
        case (crash i ls)
        then show ?thesis
          using IH2 l1 l2 by (auto simp add:  split: if_splits)
      next
        case (invCheck res i)
        then show ?thesis
          using IH2 l1 l2 by (auto simp add:  split: if_splits)
      qed

      thus ?thesis
        using  IH1' \<open>action_outputs (get_action a) \<subseteq> trace_inputs tr i1\<close> by (auto simp add: trace_outputs_append True subsetD trace_outputs_cons trace_outputs_empty)
    qed
  qed
  thus "trace_outputs (tr1 @ [a]) i1 \<subseteq> trace_inputs tr1 i1 \<union> {}"
    by (simp add: tr1a_def)
qed

end


theory mailbox_manage_sys
imports rg_cond_post func_cor_other  func_cor_OSMboxAccept  func_cor_OSMboxPost
begin

definition OSMboxPost_RGF :: "Thread \<Rightarrow> mailbox_ref \<Rightarrow> Message option \<Rightarrow> (EventLabel, Core, State, State com option) rgformula_e"
where "OSMboxPost_RGF t pevent msg  \<equiv> (OSMboxPost t pevent msg, OSMboxPost_RGCond t)"

definition OSMboxAccept_RGF :: "Thread \<Rightarrow> mailbox_ref \<Rightarrow> (EventLabel, Core, State, State com option) rgformula_e"
 where "OSMboxAccept_RGF t pevent \<equiv> (OSMboxAccept t pevent, OSMboxAccept_RGCond t)"

definition Schedule_RGF :: "Thread \<Rightarrow> (EventLabel, Core, State, State com option) rgformula_e"
where "Schedule_RGF t \<equiv> (Schedule t, Schedule_RGCond t)"

definition Tick_RGF :: "(EventLabel, Core, State, State com option) rgformula_e"
where "Tick_RGF \<equiv> (Tick, Tick_RGCond)"

definition Thread_RGF :: "Thread \<Rightarrow> (EventLabel, Core, State, State com option) rgformula_es"
where "Thread_RGF t \<equiv>  (rgf_EvtSys ((\<Union>(pevent, msg).{OSMboxPost_RGF t pevent msg}) \<union>
                         (\<Union>(pevent). {OSMboxAccept_RGF t pevent})),
         RG[(OSMboxPost_pre t \<inter> OSMboxAccept_pre t \<inter> OSMboxPend_pre t),
            (OSMboxPost_rely t \<inter> OSMboxAccept_rely t \<inter> OSMboxPend_rely t ),
            (OSMboxPost_guar t \<union> OSMboxAccept_guar t \<union> OSMboxPend_guar t),
            (OSMboxPost_post t \<union> OSMboxAccept_post t \<union> OSMboxPend_post t )
           ])"

definition Scheduler_RGF :: "(EventLabel, Core, State, State com option) rgformula_es"
where "Scheduler_RGF \<equiv>  (rgf_EvtSys (\<Union>t. {Schedule_RGF t}),
         RG[{s. inv s}, Schedule_rely, Schedule_guar, {s. inv s}])"

definition Timer_RGF :: "(EventLabel, Core, State, State com option) rgformula_es"
where "Timer_RGF \<equiv>  (rgf_EvtSys {Tick_RGF},
         RG[\<lbrace>True\<rbrace>, Tick_rely, Tick_guar, \<lbrace>True\<rbrace>])"

definition Mailbox_manage_system_Spec :: "(EventLabel, Core, State, State com option) rgformula_par"
where "Mailbox_manage_system_Spec k \<equiv>
    case k of (\<T> t) \<Rightarrow> Thread_RGF t
            | \<S> \<Rightarrow> Scheduler_RGF
            | Timer \<Rightarrow> Timer_RGF"

section \<open> functional correctness of the whole specification\<close>
definition "sys_rely \<equiv> {}"
(*definition "sys_rely \<equiv> {(s,t). inv s \<longrightarrow> inv t}"*)
(*definition "sys_rely \<equiv> (\<Inter>t. lvars_nochange_rel t) \<inter> gvars_conf_stable \<inter> {(s,t). inv s \<longrightarrow> inv t}"*)

definition "sys_guar \<equiv> Tick_guar \<union> Schedule_guar \<union> (\<Union>t. (OSMboxPost_guar t \<union> OSMboxAccept_guar t \<union> OSMboxPend_guar t ))"

lemma scheduler_esys_sat: "\<Gamma> \<turnstile> fst (Mailbox_manage_system_Spec \<S>)
  sat\<^sub>s [Pre\<^sub>e\<^sub>s (Mailbox_manage_system_Spec \<S>),
        Rely\<^sub>e\<^sub>s (Mailbox_manage_system_Spec \<S>),
        Guar\<^sub>e\<^sub>s (Mailbox_manage_system_Spec \<S>),
        Post\<^sub>e\<^sub>s (Mailbox_manage_system_Spec \<S>)]"
apply(simp add:Mailbox_manage_system_Spec_def Scheduler_RGF_def Schedule_RGF_def)
  apply(rule EvtSys_h)
    apply auto[1] apply(simp add:E\<^sub>e_def Pre\<^sub>e_def Rely\<^sub>e_def Guar\<^sub>e_def Post\<^sub>e_def)
    using Schedule_satRG apply(simp add:Schedule_RGCond_def Evt_sat_RG_def Pre\<^sub>f_def Rely\<^sub>f_def Guar\<^sub>f_def Post\<^sub>f_def)
    apply fast
    apply(simp add:Pre\<^sub>e\<^sub>s_def Pre\<^sub>e_def Schedule_RGCond_def)
    apply(simp add:Rely\<^sub>e\<^sub>s_def Rely\<^sub>e_def Schedule_RGCond_def)
    apply(simp add:Guar\<^sub>e\<^sub>s_def Guar\<^sub>e_def Schedule_RGCond_def)
    apply(simp add:Post\<^sub>e\<^sub>s_def Post\<^sub>e_def Schedule_RGCond_def)
    apply(simp add:Post\<^sub>e_def Pre\<^sub>e_def Schedule_RGCond_def getrgformula_def)
    apply(simp add:Pre\<^sub>e\<^sub>s_def Rely\<^sub>e\<^sub>s_def getrgformula_def)
      using stable_inv_sched_rely apply(simp add:stable_def)
    apply(simp add:Guar\<^sub>e\<^sub>s_def getrgformula_def Schedule_guar_def)
      done

lemma Post_rely_eq_Accept_rely:"OSMboxPost_rely x  = OSMboxAccept_rely x "
  apply(simp add:OSMboxPost_rely_def OSMboxAccept_rely_def)
  done

lemma temp:"OSMboxPost_rely x \<inter> OSMboxAccept_rely x = OSMboxPost_rely x"
  apply (simp add:OSMboxPost_rely_def OSMboxAccept_rely_def)
  done

lemma thread_esys_sat: "\<Gamma> \<turnstile> fst (Mailbox_manage_system_Spec (\<T> x))
  sat\<^sub>s [Pre\<^sub>e\<^sub>s (Mailbox_manage_system_Spec (\<T> x)),
        Rely\<^sub>e\<^sub>s (Mailbox_manage_system_Spec (\<T> x)),
        Guar\<^sub>e\<^sub>s (Mailbox_manage_system_Spec (\<T> x)),
        Post\<^sub>e\<^sub>s (Mailbox_manage_system_Spec (\<T> x))]"
apply(simp add:Mailbox_manage_system_Spec_def Thread_RGF_def OSMboxPost_RGF_def OSMboxAccept_RGF_def)
apply(rule EvtSys_h)
  apply auto[1]
    apply(simp add:E\<^sub>e_def Pre\<^sub>e_def Rely\<^sub>e_def Guar\<^sub>e_def Post\<^sub>e_def getrgformula_def OSMboxPost_RGCond_def)
      using OSMboxPost_satRG apply(simp add:Evt_sat_RG_def OSMboxPost_RGCond_def
                                      getrgformula_def Pre\<^sub>f_def Rely\<^sub>f_def Guar\<^sub>f_def Post\<^sub>f_def) apply fast
    apply(simp add:E\<^sub>e_def Pre\<^sub>e_def Rely\<^sub>e_def Guar\<^sub>e_def Post\<^sub>e_def getrgformula_def OSMboxAccept_RGCond_def)
      using OSMboxAccept_satRG apply(simp add:Evt_sat_RG_def OSMboxAccept_RGCond_def
                                      getrgformula_def Pre\<^sub>f_def Rely\<^sub>f_def Guar\<^sub>f_def Post\<^sub>f_def) apply fast
  apply auto[1]
    apply(simp add:Pre\<^sub>e\<^sub>s_def Pre\<^sub>e_def getrgformula_def OSMboxPost_RGCond_def)
    apply(simp add:Pre\<^sub>e\<^sub>s_def Pre\<^sub>e_def getrgformula_def OSMboxAccept_RGCond_def)
  apply auto[1]
    apply(simp add:Rely\<^sub>e\<^sub>s_def Rely\<^sub>e_def getrgformula_def OSMboxPost_RGCond_def)
    apply(simp add:Rely\<^sub>e\<^sub>s_def Rely\<^sub>e_def getrgformula_def OSMboxAccept_RGCond_def)
  apply auto[1]
    apply(simp add:Guar\<^sub>e\<^sub>s_def Guar\<^sub>e_def getrgformula_def OSMboxPost_RGCond_def)
    apply(simp add:Guar\<^sub>e\<^sub>s_def Guar\<^sub>e_def getrgformula_def OSMboxAccept_RGCond_def)
  apply auto[1]
    apply(simp add:Post\<^sub>e\<^sub>s_def Post\<^sub>e_def getrgformula_def OSMboxPost_RGCond_def)
    apply(simp add:Post\<^sub>e\<^sub>s_def Post\<^sub>e_def getrgformula_def OSMboxAccept_RGCond_def)
  apply auto[1]
           apply(simp add:Post\<^sub>e_def Pre\<^sub>e_def OSMboxPost_RGCond_def getrgformula_def OSMboxPost_post_def)apply auto
      apply(simp add:inv_def OSMboxPost_pre_def)
    apply(simp add:Post\<^sub>e_def Pre\<^sub>e_def OSMboxPost_RGCond_def OSMboxAccept_RGCond_def
          getrgformula_def OSMboxPost_post_def)
 apply(simp add:inv_def OSMboxAccept_pre_def)
    apply(simp add:Post\<^sub>e_def Pre\<^sub>e_def OSMboxPost_RGCond_def OSMboxAccept_RGCond_def
          getrgformula_def OSMboxAccept_post_def)
         apply(simp add:inv_def OSMboxPost_pre_def)
        apply(simp add:Post\<^sub>e_def Pre\<^sub>e_def OSMboxAccept_RGCond_def getrgformula_def OSMboxAccept_post_def)
 apply(simp add:inv_def OSMboxAccept_pre_def)
  apply(simp add:Pre\<^sub>e\<^sub>s_def Rely\<^sub>e\<^sub>s_def getrgformula_def)
    defer 1 (* proved in mem_alloc *)
       apply(simp add:Guar\<^sub>e\<^sub>s_def getrgformula_def OSMboxAccept_guar_def)

      apply(simp add:OSMboxAccept_pre_def OSMboxPost_pre_def OSMboxPend_pre_def)
      apply(simp add:temp)
      apply(simp add:inv_def OSMboxPost_rely_def inv_cur_def stable_def OSMboxPend_rely_def)
      done

lemma timer_esys_sat: "\<Gamma> \<turnstile> fst (Mailbox_manage_system_Spec Timer)
  sat\<^sub>s [Pre\<^sub>e\<^sub>s (Mailbox_manage_system_Spec Timer),
        Rely\<^sub>e\<^sub>s (Mailbox_manage_system_Spec Timer),
        Guar\<^sub>e\<^sub>s (Mailbox_manage_system_Spec Timer),
        Post\<^sub>e\<^sub>s (Mailbox_manage_system_Spec Timer)]"
apply(simp add:Mailbox_manage_system_Spec_def Timer_RGF_def Tick_RGF_def)
apply(rule EvtSys_h)
  apply auto[1] apply(simp add:E\<^sub>e_def Pre\<^sub>e_def Rely\<^sub>e_def Guar\<^sub>e_def Post\<^sub>e_def)
    using Tick_satRG apply(simp add:Tick_RGCond_def Evt_sat_RG_def Pre\<^sub>f_def Rely\<^sub>f_def Guar\<^sub>f_def Post\<^sub>f_def)
    apply fast
  apply(simp add:Pre\<^sub>e\<^sub>s_def Pre\<^sub>e_def Tick_RGCond_def)
  apply(simp add:Rely\<^sub>e\<^sub>s_def Rely\<^sub>e_def Tick_RGCond_def)
  apply(simp add:Guar\<^sub>e\<^sub>s_def Guar\<^sub>e_def Tick_RGCond_def)
  apply(simp add:Post\<^sub>e\<^sub>s_def Post\<^sub>e_def Tick_RGCond_def)
  apply(simp add:Post\<^sub>e_def Pre\<^sub>e_def Tick_RGCond_def getrgformula_def)
  apply(simp add:Pre\<^sub>e\<^sub>s_def Rely\<^sub>e\<^sub>s_def getrgformula_def)
    using stable_inv_sched_rely apply(simp add:stable_def)
  apply(simp add:Guar\<^sub>e\<^sub>s_def getrgformula_def Tick_guar_def)
done

lemma esys_sat: "\<Gamma> \<turnstile> fst (Mailbox_manage_system_Spec k)
  sat\<^sub>s [Pre\<^sub>e\<^sub>s (Mailbox_manage_system_Spec k),
        Rely\<^sub>e\<^sub>s (Mailbox_manage_system_Spec k),
        Guar\<^sub>e\<^sub>s (Mailbox_manage_system_Spec k),
        Post\<^sub>e\<^sub>s (Mailbox_manage_system_Spec k)]"
  apply(induct k)
  using scheduler_esys_sat apply fast
  using thread_esys_sat apply fast
  using timer_esys_sat apply fast
done

lemma s0_esys_pre: "{s0} \<subseteq> Pre\<^sub>e\<^sub>s (Mailbox_manage_system_Spec k)"
apply(induct k)
  apply(simp add:Mailbox_manage_system_Spec_def Pre\<^sub>e\<^sub>s_def Scheduler_RGF_def getrgformula_def)    (*System 1: Scheduler *)
  using s0_inv apply fast

     apply(simp add:Mailbox_manage_system_Spec_def Pre\<^sub>e\<^sub>s_def Thread_RGF_def getrgformula_def)    (*System 2: Thread *)
    using s0_inv s0a4 s0a1 s0a2 s0a3 apply auto[1]
      apply(simp add:inv_def OSMboxPost_pre_def)
      apply(simp add:inv_def OSMboxAccept_pre_def)
     apply(simp add:inv_def OSMboxPend_pre_def)

  apply(simp add:Mailbox_manage_system_Spec_def Pre\<^sub>e\<^sub>s_def Timer_RGF_def getrgformula_def)       (*System 3: Timer *)
done

lemma alloc_free_eq_guar: " OSMboxPost_guar x\<subseteq> OSMboxAccept_guar x "
  apply(simp add:OSMboxAccept_guar_def OSMboxPost_guar_def)
  apply(simp add:gvars_conf_stable_def)
  apply auto
         apply(simp add: lvars_nochange_def lvars_nochange_lemma_def gvars_conf_stable_def gvars_conf_def)apply auto

lemma alloc_free_eq_rely: "OSMboxAccept_rely x = OSMboxPost_rely x"
  by(simp add:OSMboxAccept_rely_def OSMboxPost_rely_def)

lemma esys_guar_in_other:
  "jj \<noteq> k \<longrightarrow> Guar\<^sub>e\<^sub>s (Mailbox_manage_system_Spec jj) \<subseteq> Rely\<^sub>e\<^sub>s (Mailbox_manage_system_Spec k)"
  apply auto
  apply(induct jj)
    apply(induct k)
      apply simp
     apply(simp add:Guar\<^sub>e\<^sub>s_def Rely\<^sub>e\<^sub>s_def Mailbox_manage_system_Spec_def Scheduler_RGF_def Thread_RGF_def getrgformula_def)
  using Schedguar_in_OSMemGetrely  apply(simp add:OSMboxPost_rely_def OSMboxAccept_rely_def) apply auto[1]
  apply(induct k)
  using OSMemGetguar_in_OSMemGetrely OSMem_eq_guar OSMem_eq_rely by fast+

thm esys_guar_in_other

lemma esys_guar_in_sys: "Guar\<^sub>e\<^sub>s (Mailbox_manage_system_Spec k) \<subseteq> sys_guar"
apply(induct k)
  apply(simp add:Guar\<^sub>e\<^sub>s_def Mailbox_manage_system_Spec_def Scheduler_RGF_def getrgformula_def sys_guar_def) apply auto[1]
  apply(simp add:Guar\<^sub>e\<^sub>s_def Mailbox_manage_system_Spec_def Thread_RGF_def getrgformula_def sys_guar_def) apply auto[1]
  apply(simp add:Guar\<^sub>e\<^sub>s_def Mailbox_manage_system_Spec_def Timer_RGF_def getrgformula_def sys_guar_def) apply auto[1]
done

lemma mailbox_sys_sat: "\<Gamma> \<turnstile> Mailbox_manage_system_Spec SAT [{s0}, sys_rely, sys_guar, UNIV]"
apply(rule ParallelESys[of \<Gamma> Mailbox_manage_system_Spec"{s0}" sys_rely sys_guar UNIV])
  apply clarify using esys_sat apply fast
  using s0_esys_pre apply fast
  apply(simp add:sys_rely_def)
  using esys_guar_in_other apply fast
  using esys_guar_in_sys apply fast
  apply simp
  done

end