Added headers and made various small mods.
(* Title: HOL/IOA/example/Impl.ML
ID: $Id$
Author: Tobias Nipkow & Konrad Slind
Copyright 1994 TU Muenchen
The implementation --- Invariants
*)
val impl_ioas =
[Impl.impl_def,
Sender.sender_ioa_def,
Receiver.receiver_ioa_def,
Channels.srch_ioa_def,
Channels.rsch_ioa_def];
val transitions = [Sender.sender_trans_def, Receiver.receiver_trans_def,
Channels.srch_trans_def, Channels.rsch_trans_def];
val impl_ss = merge_ss(action_ss,list_ss)
addcongs [let_weak_cong]
addsimps [Let_def, ioa_triple_proj, starts_of_par, trans_of_par4,
in_sender_asig, in_receiver_asig, in_srch_asig,
in_rsch_asig, count_addm_simp, count_delm_simp,
Multiset.countm_empty_def, Multiset.delm_empty_def,
(* Multiset.count_def, *) count_empty,
Packet.hdr_def, Packet.msg_def];
goal Impl.thy
"fst(x) = sen(x) & \
\ fst(snd(x)) = rec(x) & \
\ fst(snd(snd(x))) = srch(x) & \
\ snd(snd(snd(x))) = rsch(x)";
by(simp_tac (HOL_ss addsimps
[Impl.sen_def,Impl.rec_def,Impl.srch_def,Impl.rsch_def]) 1);
val impl_ss = impl_ss addsimps [result()];
goal Impl.thy "a:actions(sender_asig) \
\ | a:actions(receiver_asig) \
\ | a:actions(srch_asig) \
\ | a:actions(rsch_asig)";
by(Action.action.induct_tac "a" 1);
by(ALLGOALS(simp_tac impl_ss));
val impl_ss = impl_ss addsimps [result()];
(* Instantiation of a tautology? *)
goal Packet.thy "!x. x = packet --> hdr(x) = hdr(packet)";
by (simp_tac (HOL_ss addsimps [Packet.hdr_def]) 1);
val eq_packet_imp_eq_hdr = result();
(* INVARIANT 1 *)
val ss = impl_ss addcongs [if_weak_cong] addsimps transitions;
goalw Impl.thy impl_ioas "invariant(impl_ioa,inv1)";
br invariantI 1;
by(asm_full_simp_tac (impl_ss addsimps
[Impl.inv1_def, Impl.hdr_sum_def,
Sender.srcvd_def, Sender.ssent_def,
Receiver.rsent_def,Receiver.rrcvd_def]) 1);
by(simp_tac (HOL_ss addsimps [fork_lemma,Impl.inv1_def]) 1);
(* Split proof in two *)
by (rtac conjI 1);
(* First half *)
by(asm_full_simp_tac (impl_ss addsimps [Impl.inv1_def]) 1);
br Action.action.induct 1;
val tac = asm_simp_tac (ss addcongs [conj_cong]
addsimps [Suc_pred_lemma]
setloop (split_tac [expand_if]));
by(EVERY1[tac, tac, tac, tac, tac, tac, tac, tac, tac, tac]);
(* Now the other half *)
by(asm_full_simp_tac (impl_ss addsimps [Impl.inv1_def]) 1);
br Action.action.induct 1;
by(EVERY1[tac, tac]);
(* detour 1 *)
by (tac 1);
by (rtac impI 1);
by (REPEAT (etac conjE 1));
by (asm_simp_tac (impl_ss addsimps [Impl.hdr_sum_def, Multiset.count_def,
Multiset.countm_nonempty_def]
setloop (split_tac [expand_if])) 1);
(* detour 2 *)
by (tac 1);
by (rtac impI 1);
by (REPEAT (etac conjE 1));
by (asm_full_simp_tac (impl_ss addsimps
[Impl.hdr_sum_def,
Multiset.count_def,
Multiset.countm_nonempty_def,
Multiset.delm_nonempty_def,
left_plus_cancel,left_plus_cancel_inside_succ,
unzero_less]
setloop (split_tac [expand_if])) 1);
by (rtac allI 1);
by (rtac conjI 1);
by (rtac impI 1);
by (hyp_subst_tac 1);
by (rtac (pred_suc RS mp RS sym RS iffD2) 1);
by (dmatch_tac [less_leq_less RS mp] 1);
by (cut_facts_tac [rewrite_rule[Packet.hdr_def]
eq_packet_imp_eq_hdr RS countm_props] 1);;
by (dtac mp 1);
by (assume_tac 1);
by (assume_tac 1);
by (rtac (countm_done_delm RS mp RS sym) 1);
by (rtac refl 1);
by (asm_simp_tac (HOL_ss addsimps [Multiset.count_def]) 1);
by (rtac impI 1);
by (asm_full_simp_tac (HOL_ss addsimps [neg_flip]) 1);
by (hyp_subst_tac 1);
by (rtac countm_spurious_delm 1);
by (simp_tac HOL_ss 1);
by (EVERY1[tac, tac, tac, tac, tac, tac]);
val inv1 = result();
(* INVARIANT 2 *)
goal Impl.thy "invariant(impl_ioa, inv2)";
by (rtac invariantI1 1);
(* Base case *)
by (asm_full_simp_tac (impl_ss addsimps
(Impl.inv2_def :: (receiver_projections
@ sender_projections @ impl_ioas))) 1);
by (asm_simp_tac (impl_ss addsimps impl_ioas) 1);
by (Action.action.induct_tac "a" 1);
(* 10 cases. First 4 are simple, since state doesn't change wrt. invariant *)
(* 10 *)
by (asm_simp_tac (impl_ss addsimps (Impl.inv2_def::transitions)) 1);
(* 9 *)
by (asm_simp_tac (impl_ss addsimps (Impl.inv2_def::transitions)) 1);
(* 8 *)
by (asm_simp_tac (impl_ss addsimps (Impl.inv2_def::transitions)) 2);
(* 7 *)
by (asm_simp_tac (impl_ss addsimps (Impl.inv2_def::transitions)) 3);
(* 6 *)
by(forward_tac [rewrite_rule [Impl.inv1_def]
(inv1 RS invariantE) RS conjunct1] 1);
by (asm_full_simp_tac (impl_ss addsimps [leq_imp_leq_suc,Impl.inv2_def]
addsimps transitions) 1);
(* 5 *)
by (asm_full_simp_tac (impl_ss addsimps [leq_imp_leq_suc,Impl.inv2_def]
addsimps transitions) 1);
(* 4 *)
by (forward_tac [rewrite_rule [Impl.inv1_def]
(inv1 RS invariantE) RS conjunct1] 1);
by (asm_full_simp_tac (impl_ss addsimps [Impl.inv2_def]
addsimps transitions) 1);
by (fast_tac (HOL_cs addDs [plus_leD1,leD]) 1);
(* 3 *)
by (forward_tac [rewrite_rule [Impl.inv1_def] (inv1 RS invariantE)] 1);
by (asm_full_simp_tac (impl_ss addsimps
(Impl.inv2_def::transitions)) 1);
by (fold_tac [rewrite_rule [Packet.hdr_def]Impl.hdr_sum_def]);
by (fast_tac (HOL_cs addDs [plus_leD1,leD]) 1);
(* 2 *)
by (asm_full_simp_tac (impl_ss addsimps
(Impl.inv2_def::transitions)) 1);
by(forward_tac [rewrite_rule [Impl.inv1_def]
(inv1 RS invariantE) RS conjunct1] 1);
by (rtac impI 1);
by (rtac impI 1);
by (REPEAT (etac conjE 1));
by (dres_inst_tac [("k","count(rsch(s), ~ sbit(sen(s)))")]
(standard(leq_add_leq RS mp)) 1);
by (asm_full_simp_tac HOL_ss 1);
(* 1 *)
by (asm_full_simp_tac (impl_ss addsimps
(Impl.inv2_def::transitions)) 1);
by(forward_tac [rewrite_rule [Impl.inv1_def]
(inv1 RS invariantE) RS conjunct2] 1);
by (rtac impI 1);
by (rtac impI 1);
by (REPEAT (etac conjE 1));
by (fold_tac [rewrite_rule[Packet.hdr_def]Impl.hdr_sum_def]);
by (dres_inst_tac [("k","hdr_sum(srch(s), sbit(sen(s)))")]
(standard(leq_add_leq RS mp)) 1);
by (asm_full_simp_tac HOL_ss 1);
val inv2 = result();
(* INVARIANT 3 *)
goal Impl.thy "invariant(impl_ioa, inv3)";
by (rtac invariantI 1);
(* Base case *)
by (asm_full_simp_tac (impl_ss addsimps
(Impl.inv3_def :: (receiver_projections
@ sender_projections @ impl_ioas))) 1);
by (asm_simp_tac (impl_ss addsimps impl_ioas) 1);
by (Action.action.induct_tac "a" 1);
(* 10 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::not_hd_append::Impl.inv3_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 9 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::not_hd_append::Impl.inv3_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 8 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::not_hd_append::Impl.inv3_def::transitions)
setloop (split_tac [expand_if])) 1);
by (strip_tac 1 THEN REPEAT (etac conjE 1));
by (asm_full_simp_tac (HOL_ss addsimps [cons_imp_not_null]) 1);
by (hyp_subst_tac 1);
by (etac exE 1);
by (asm_full_simp_tac list_ss 1);
(* 7 *)
by (asm_full_simp_tac (impl_ss addsimps
(Suc_pred_lemma::append_cons::not_hd_append::Impl.inv3_def::transitions)
setloop (split_tac [expand_if])) 1);
by (fast_tac HOL_cs 1);
(* 6 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::not_hd_append::Impl.inv3_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 5 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::not_hd_append::Impl.inv3_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 4 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::not_hd_append::Impl.inv3_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 3 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::not_hd_append::Impl.inv3_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 2 *)
by (asm_full_simp_tac (impl_ss addsimps transitions) 1);
by (simp_tac (HOL_ss addsimps [Impl.inv3_def]) 1);
by (strip_tac 1 THEN REPEAT (etac conjE 1));
by (rtac (imp_or_lem RS iffD2) 1);
by (rtac impI 1);
by (forward_tac [rewrite_rule [Impl.inv2_def] (inv2 RS invariantE)] 1);
by (asm_full_simp_tac list_ss 1);
by (REPEAT (etac conjE 1));
by (res_inst_tac [("j","count(ssent(sen(s)),~ sbit(sen(s)))"),
("k","count(rsent(rec(s)), sbit(sen(s)))")] le_trans 1);
by (forward_tac [rewrite_rule [Impl.inv1_def]
(inv1 RS invariantE) RS conjunct2] 1);
by (asm_full_simp_tac (list_ss addsimps
[Impl.hdr_sum_def, Multiset.count_def]) 1);
by (rtac (less_eq_add_cong RS mp RS mp) 1);
by (rtac countm_props 1);
by (simp_tac (list_ss addsimps [Packet.hdr_def]) 1);
by (rtac countm_props 1);
by (simp_tac (list_ss addsimps [Packet.hdr_def]) 1);
by (assume_tac 1);
(* 1 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::not_hd_append::Impl.inv3_def::transitions)
setloop (split_tac [expand_if])) 1);
by (strip_tac 1 THEN REPEAT (etac conjE 1));
by (rtac (imp_or_lem RS iffD2) 1);
by (rtac impI 1);
by (forward_tac [rewrite_rule [Impl.inv2_def] (inv2 RS invariantE)] 1);
by (asm_full_simp_tac list_ss 1);
by (REPEAT (etac conjE 1));
by (dtac mp 1);
by (assume_tac 1);
by (etac allE 1);
by (dtac (imp_or_lem RS iffD1) 1);
by (dtac mp 1);
by (assume_tac 1);
by (assume_tac 1);
val inv3 = result();
(* INVARIANT 4 *)
goal Impl.thy "invariant(impl_ioa, inv4)";
by (rtac invariantI 1);
(* Base case *)
by (asm_full_simp_tac (impl_ss addsimps
(Impl.inv4_def :: (receiver_projections
@ sender_projections @ impl_ioas))) 1);
by (asm_simp_tac (impl_ss addsimps impl_ioas) 1);
by (Action.action.induct_tac "a" 1);
(* 10 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::Impl.inv4_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 9 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::Impl.inv4_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 8 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::Impl.inv4_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 7 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::Impl.inv4_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 6 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::Impl.inv4_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 5 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::Impl.inv4_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 4 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::Impl.inv4_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 3 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::Impl.inv4_def::transitions)
setloop (split_tac [expand_if])) 1);
(* 2 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::Impl.inv4_def::transitions)
setloop (split_tac [expand_if])) 1);
by (strip_tac 1 THEN REPEAT (etac conjE 1));
by(forward_tac [rewrite_rule [Impl.inv2_def]
(inv2 RS invariantE)] 1);
by (asm_full_simp_tac list_ss 1);
(* 1 *)
by (asm_full_simp_tac (impl_ss addsimps
(append_cons::Impl.inv4_def::transitions)
setloop (split_tac [expand_if])) 1);
by (strip_tac 1 THEN REPEAT (etac conjE 1));
by (rtac ccontr 1);
by(forward_tac [rewrite_rule [Impl.inv2_def]
(inv2 RS invariantE)] 1);
by(forward_tac [rewrite_rule [Impl.inv3_def]
(inv3 RS invariantE)] 1);
by (asm_full_simp_tac list_ss 1);
by (eres_inst_tac [("x","m")] allE 1);
by (dtac (less_leq_less RS mp RS mp) 1);
by (dtac (left_add_leq RS mp) 1);
by (asm_full_simp_tac list_ss 1);
by (asm_full_simp_tac arith_ss 1);
val inv4 = result();
(* rebind them *)
val inv1 = rewrite_rule [Impl.inv1_def] (inv1 RS invariantE);
val inv2 = rewrite_rule [Impl.inv2_def] (inv2 RS invariantE);
val inv3 = rewrite_rule [Impl.inv3_def] (inv3 RS invariantE);
val inv4 = rewrite_rule [Impl.inv4_def] (inv4 RS invariantE);