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(* Title: HOL/Quot/PER0.ML
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ID: $Id$
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Author: Oscar Slotosch
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Copyright 1997 Technische Universitaet Muenchen
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*)
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open PER0;
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(* derive the characteristic axioms *)
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val prems = goalw thy [per_def] "x === y ==> y === x";
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by (cut_facts_tac prems 1);
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be ax_per_sym 1;
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qed "per_sym";
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val prems = goalw thy [per_def] "[| x === y; y === z |] ==> x === z";
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by (cut_facts_tac prems 1);
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be ax_per_trans 1;
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ba 1;
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qed "per_trans";
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goalw thy [per_def] "(x::'a::er) === x";
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br ax_er_refl 1;
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qed "er_refl";
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(* now everything works without axclasses *)
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goal thy "x===y=y===x";
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br iffI 1;
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be per_sym 1;
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be per_sym 1;
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qed "per_sym2";
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val prems = goal thy "x===y ==> x===x";
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by (cut_facts_tac prems 1);
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br per_trans 1;ba 1;
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be per_sym 1;
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qed "sym2refl1";
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val prems = goal thy "x===y ==> y===y";
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by (cut_facts_tac prems 1);
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br per_trans 1;ba 2;
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be per_sym 1;
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qed "sym2refl2";
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val prems = goalw thy [Dom] "x:D ==> x === x";
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by (cut_facts_tac prems 1);
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by (fast_tac set_cs 1);
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qed "DomainD";
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val prems = goalw thy [Dom] "x === x ==> x:D";
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by (cut_facts_tac prems 1);
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by (fast_tac set_cs 1);
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qed "DomainI";
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goal thy "x:D = x===x";
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br iffI 1;
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be DomainD 1;
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be DomainI 1;
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qed "DomainEq";
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goal thy "(~ x === y) = (~ y === x)";
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br (per_sym2 RS arg_cong) 1;
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qed "per_not_sym";
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(* show that PER work only on D *)
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val prems = goal thy "x===y ==> x:D";
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by (cut_facts_tac prems 1);
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be (sym2refl1 RS DomainI) 1;
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qed "DomainI_left";
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val prems = goal thy "x===y ==> y:D";
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by (cut_facts_tac prems 1);
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be (sym2refl2 RS DomainI) 1;
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qed "DomainI_right";
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val prems = goalw thy [Dom] "x~:D ==> ~x===y";
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by (cut_facts_tac prems 1);
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by (res_inst_tac [("Q","x===y")] (excluded_middle RS disjE) 1);ba 1;
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bd sym2refl1 1;
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by (fast_tac set_cs 1);
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qed "notDomainE1";
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val prems = goalw thy [Dom] "x~:D ==> ~y===x";
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by (cut_facts_tac prems 1);
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by (res_inst_tac [("Q","y===x")] (excluded_middle RS disjE) 1);ba 1;
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bd sym2refl2 1;
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by (fast_tac set_cs 1);
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qed "notDomainE2";
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(* theorems for equivalence relations *)
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goal thy "(x::'a::er) : D";
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br DomainI 1;
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br er_refl 1;
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qed "er_Domain";
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(* witnesses for "=>" ::(per,per)per *)
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val prems = goalw thy [fun_per_def]"eqv (x::'a::per => 'b::per) y ==> eqv y x";
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by (cut_facts_tac prems 1);
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by (safe_tac HOL_cs);
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by (asm_full_simp_tac (HOL_ss addsimps [per_sym2]) 1);
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qed "per_sym_fun";
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val prems = goalw thy [fun_per_def]
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"[| eqv (f::'a::per=>'b::per) g;eqv g h|] ==> eqv f h";
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by (cut_facts_tac prems 1);
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by (safe_tac HOL_cs);
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by (REPEAT (dtac spec 1));
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by (res_inst_tac [("y","g y")] per_trans 1);
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br mp 1;ba 1;
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by (Asm_simp_tac 1);
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br mp 1;ba 1;
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by (asm_simp_tac (!simpset addsimps [sym2refl2]) 1);
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qed "per_trans_fun";
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