(* Title: HOL/UNITY/Token
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
The Token Ring.
From Misra, "A Logic for Concurrent Programming" (1994), sections 5.2 and 13.2.
*)
val Token_defs = [HasTok_def, H_def, E_def, T_def];
Goalw [HasTok_def] "[| s: HasTok i; s: HasTok j |] ==> i=j";
by Auto_tac;
qed "HasToK_partition";
Goalw Token_defs "(s ~: E i) = (s : H i | s : T i)";
by (Simp_tac 1);
by (exhaust_tac "proc s i" 1);
by Auto_tac;
qed "not_E_eq";
Open_locale "Token";
(*Strip meta-quantifiers: perhaps the locale should do this?*)
val TR2 = forall_elim_vars 0 (thm "TR2");
val TR3 = forall_elim_vars 0 (thm "TR3");
val TR4 = forall_elim_vars 0 (thm "TR4");
val TR5 = forall_elim_vars 0 (thm "TR5");
val TR6 = forall_elim_vars 0 (thm "TR6");
val TR7 = forall_elim_vars 0 (thm "TR7");
val nodeOrder_def = (thm "nodeOrder_def");
val next_def = (thm "next_def");
AddIffs [thm "N_positive"];
Goalw [stable_def] "F : stable (-(E i) Un (HasTok i))";
by (rtac constrains_weaken 1);
by (rtac ([[TR2, TR4] MRS constrains_Un, TR5] MRS constrains_Un) 1);
by (auto_tac (claset(), simpset() addsimps [not_E_eq]));
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [H_def, E_def, T_def])));
qed "token_stable";
(*** Progress under weak fairness ***)
Goalw [nodeOrder_def] "wf(nodeOrder j)";
by (rtac (wf_less_than RS wf_inv_image RS wf_subset) 1);
by (Blast_tac 1);
qed"wf_nodeOrder";
Goalw [nodeOrder_def, next_def, inv_image_def]
"[| i<N; j<N |] ==> ((next i, i) : nodeOrder j) = (i ~= j)";
by (auto_tac (claset(),
simpset() addsimps [Suc_lessI, mod_Suc, mod_less, mod_geq]));
by (subgoal_tac "(j + N - i) = Suc (j + N - Suc i)" 1);
by (asm_simp_tac (simpset() addsimps [Suc_diff_Suc, Suc_leI, less_imp_le,
diff_add_assoc]) 2);
by (full_simp_tac (simpset() addsimps [linorder_neq_iff]) 1);
by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq, mod_Suc]) 1);
by (auto_tac (claset(),
simpset() addsimps [diff_add_assoc2, linorder_neq_iff,
Suc_le_eq, Suc_diff_Suc, less_imp_diff_less,
add_diff_less, mod_less, mod_geq]));
by (etac subst 1);
by (asm_simp_tac (simpset() addsimps [add_diff_less]) 1);
qed "nodeOrder_eq";
(*From "A Logic for Concurrent Programming", but not used in Chapter 4.
Note the use of case_tac. Reasoning about leadsTo takes practice!*)
Goal "[| i<N; j<N |] ==> \
\ F : leadsTo (HasTok i) ({s. (token s, i) : nodeOrder j} Un HasTok j)";
by (case_tac "i=j" 1);
by (blast_tac (claset() addIs [subset_imp_leadsTo]) 1);
by (rtac (TR7 RS leadsTo_weaken_R) 1);
by (Clarify_tac 1);
by (asm_full_simp_tac (simpset() addsimps [HasTok_def, nodeOrder_eq]) 1);
qed "TR7_nodeOrder";
(*Chapter 4 variant, the one actually used below.*)
Goal "[| i<N; j<N; i~=j |] \
\ ==> F : leadsTo (HasTok i) {s. (token s, i) : nodeOrder j}";
by (rtac (TR7 RS leadsTo_weaken_R) 1);
by (Clarify_tac 1);
by (asm_full_simp_tac (simpset() addsimps [HasTok_def, nodeOrder_eq]) 1);
qed "TR7_aux";
Goal "({s. token s < N} Int token -`` {m}) = \
\ (if m<N then token -`` {m} else {})";
by Auto_tac;
val token_lemma = result();
(*Misra's TR9: the token reaches an arbitrary node*)
Goal "j<N ==> F : leadsTo {s. token s < N} (HasTok j)";
by (rtac leadsTo_weaken_R 1);
by (res_inst_tac [("I", "-{j}"), ("f", "token"), ("B", "{}")]
(wf_nodeOrder RS bounded_induct) 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [token_lemma, vimage_Diff,
HasTok_def])));
by (Blast_tac 2);
by (Clarify_tac 1);
by (rtac (TR7_aux RS leadsTo_weaken) 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [nodeOrder_def, HasTok_def])));
by (ALLGOALS Blast_tac);
qed "leadsTo_j";
(*Misra's TR8: a hungry process eventually eats*)
Goal "j<N ==> F : leadsTo ({s. token s < N} Int H j) (E j)";
by (rtac (leadsTo_cancel1 RS leadsTo_Un_duplicate) 1);
by (rtac TR6 2);
by (rtac leadsTo_weaken_R 1);
by (rtac ([leadsTo_j, TR3] MRS psp) 1);
by (ALLGOALS Blast_tac);
qed "token_progress";