4776
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(* Title: HOL/UNITY/Channel
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1998 University of Cambridge
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Unordered Channel
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From Misra, "A Logic for Concurrent Programming" (1994), section 13.3
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*)
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(*None represents "infinity" while Some represents proper integers*)
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5111
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Goalw [minSet_def] "minSet A = Some x --> x : A";
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4776
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by (Simp_tac 1);
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by (fast_tac (claset() addIs [LeastI]) 1);
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qed_spec_mp "minSet_eq_SomeD";
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5069
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Goalw [minSet_def] " minSet{} = None";
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4776
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by (Asm_simp_tac 1);
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qed_spec_mp "minSet_empty";
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Addsimps [minSet_empty];
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5111
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Goalw [minSet_def] "x:A ==> minSet A = Some (LEAST x. x: A)";
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7403
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by Auto_tac;
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4776
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qed_spec_mp "minSet_nonempty";
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6536
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Goal "F : (minSet -`` {Some x}) leadsTo (minSet -`` (Some``greaterThan x))";
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4776
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by (rtac leadsTo_weaken 1);
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6700
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by (res_inst_tac [("x1","x")] ([UC2, UC1] MRS psp) 1);
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4776
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by Safe_tac;
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by (auto_tac (claset() addDs [minSet_eq_SomeD],
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6700
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simpset() addsimps [linorder_neq_iff]));
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4776
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qed "minSet_greaterThan";
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(*The induction*)
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6536
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Goal "F : (UNIV-{{}}) leadsTo (minSet -`` (Some``atLeast y))";
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4776
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by (rtac leadsTo_weaken_R 1);
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by (res_inst_tac [("l", "y"), ("f", "the o minSet"), ("B", "{}")]
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greaterThan_bounded_induct 1);
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by Safe_tac;
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by (ALLGOALS Asm_simp_tac);
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by (dtac minSet_nonempty 2);
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by (Asm_full_simp_tac 2);
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by (rtac (minSet_greaterThan RS leadsTo_weaken) 1);
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by Safe_tac;
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by (ALLGOALS Asm_full_simp_tac);
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by (dtac minSet_nonempty 1);
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by (Asm_full_simp_tac 1);
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val lemma = result();
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6536
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Goal "!!y::nat. F : (UNIV-{{}}) leadsTo {s. y ~: s}";
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4776
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by (rtac (lemma RS leadsTo_weaken_R) 1);
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by (Clarify_tac 1);
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7499
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by (ftac minSet_nonempty 1);
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7403
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by (auto_tac (claset() addDs [Suc_le_lessD, not_less_Least],
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simpset()));
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4776
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qed "Channel_progress";
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