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open OrdDefs;
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(** lifting of quasi / partial orders **)
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(* pairs *)
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goalw thy [le_prod_def] "x [= (x::'a::quasi_order*'b::quasi_order)";
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br conjI 1;
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br le_refl 1;
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br le_refl 1;
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qed "le_prod_refl";
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goalw thy [le_prod_def] "x [= y & y [= z --> x [= (z::'a::quasi_order*'b::quasi_order)";
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by (safe_tac HOL_cs);
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be (conjI RS (le_trans RS mp)) 1;
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ba 1;
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be (conjI RS (le_trans RS mp)) 1;
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ba 1;
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qed "le_prod_trans";
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goalw thy [le_prod_def] "x [= y & y [= x --> x = (y::'a::order*'b::order)";
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by (safe_tac HOL_cs);
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by (stac Pair_fst_snd_eq 1);
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br conjI 1;
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be (conjI RS (le_antisym RS mp)) 1;
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ba 1;
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be (conjI RS (le_antisym RS mp)) 1;
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ba 1;
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qed "le_prod_antisym";
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(* functions *)
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goalw thy [le_fun_def] "f [= (f::'a=>'b::quasi_order)";
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br allI 1;
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br le_refl 1;
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qed "le_fun_refl";
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goalw thy [le_fun_def] "f [= g & g [= h --> f [= (h::'a=>'b::quasi_order)";
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by (safe_tac HOL_cs);
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br (le_trans RS mp) 1;
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by (fast_tac HOL_cs 1);
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qed "le_fun_trans";
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goalw thy [le_fun_def] "f [= g & g [= f --> f = (g::'a=>'b::order)";
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by (safe_tac HOL_cs);
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br ext 1;
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br (le_antisym RS mp) 1;
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by (fast_tac HOL_cs 1);
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qed "le_fun_antisym";
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(** duals **)
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(*"'a dual" is even an isotype*)
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goal thy "Rep_dual (Abs_dual y) = y";
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br Abs_dual_inverse 1;
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1465
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by (rewtac dual_def);
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1440
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by (fast_tac set_cs 1);
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qed "Abs_dual_inverse'";
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goalw thy [le_dual_def] "x [= (x::'a::quasi_order dual)";
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br le_refl 1;
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qed "le_dual_refl";
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goalw thy [le_dual_def] "x [= y & y [= z --> x [= (z::'a::quasi_order dual)";
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by (stac conj_commut 1);
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br le_trans 1;
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qed "le_dual_trans";
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goalw thy [le_dual_def] "x [= y & y [= x --> x = (y::'a::order dual)";
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by (safe_tac prop_cs);
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br (Rep_dual_inverse RS subst) 1;
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br sym 1;
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br (Rep_dual_inverse RS subst) 1;
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br arg_cong 1;
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back();
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be (conjI RS (le_antisym RS mp)) 1;
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ba 1;
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qed "le_dual_antisym";
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goalw thy [le_dual_def] "x [= y | y [= (x::'a::lin_order dual)";
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br le_lin 1;
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qed "le_dual_lin";
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