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(* Title: Psubset.ML
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Author: Martin Coen, Cambridge University Computer Laboratory
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Copyright 1993 University of Cambridge
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Properties of subsets and empty sets.
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*)
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open Psubset;
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(*********)
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(*** Rules for subsets ***)
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goal Set.thy "A <= B = (! t.t:A --> t:B)";
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by (Blast_tac 1);
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qed "subset_iff";
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goalw thy [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
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by (Blast_tac 1);
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qed "psubsetI";
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goalw thy [psubset_def] "((A::'a set) <= B) = ((A < B) | (A=B))";
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by (Blast_tac 1);
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qed "subset_iff_psubset_eq";
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goal Set.thy "!!a. insert a A ~= insert a B ==> A ~= B";
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by (Blast_tac 1);
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qed "insert_lim";
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(* This is an adaptation of the proof for the "<=" version in Finite. *)
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goalw thy [psubset_def]
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"!!B. finite B ==> !A. A < B --> card(A) < card(B)";
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by (etac finite_induct 1);
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by (Simp_tac 1);
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by (Blast_tac 1);
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by (strip_tac 1);
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by (etac conjE 1);
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by (case_tac "x:A" 1);
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(*1*)
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by (dtac mk_disjoint_insert 1);
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by (etac exE 1);
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by (etac conjE 1);
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by (hyp_subst_tac 1);
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by (rotate_tac ~1 1);
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by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1);
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by (dtac insert_lim 1);
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by (Asm_full_simp_tac 1);
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(*2*)
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by (rotate_tac ~1 1);
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by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1);
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by (case_tac "A=F" 1);
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by (Asm_simp_tac 1);
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by (Asm_simp_tac 1);
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by (subgoal_tac "card A <= card F" 1);
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by (Asm_simp_tac 2);
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by (Auto_tac());
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qed_spec_mp "psubset_card" ;
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goal Set.thy "(A = B) = ((A <= (B::'a set)) & (B<=A))";
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by (Blast_tac 1);
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qed "set_eq_subset";
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goalw thy [psubset_def] "~ (A < {})";
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by (Blast_tac 1);
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qed "not_psubset_empty";
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AddIffs [not_psubset_empty];
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goalw thy [psubset_def]
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"!!x. A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
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by (Auto_tac());
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qed "psubset_insertD";
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(*NB we do not have [| A < B; C < D |] ==> A Un C < B Un D
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even for finite sets: consider A={1}, C={2}, B=D={1,2} *)
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