src/HOL/WF_Rel.ML
author paulson
Mon, 23 Jun 1997 10:42:03 +0200
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(*  Title: 	HOL/WF_Rel
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    ID:         $Id$
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    Author: 	Konrad Slind
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    Copyright   1996  TU Munich
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Derived WF relations: inverse image, lexicographic product, measure, ...
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*)
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open WF_Rel;
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(*----------------------------------------------------------------------------
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 * "Less than" on the natural numbers
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 *---------------------------------------------------------------------------*)
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goalw thy [less_than_def] "wf less_than"; 
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by (rtac (wf_pred_nat RS wf_trancl) 1);
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qed "wf_less_than";
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AddIffs [wf_less_than];
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goalw thy [less_than_def] "trans less_than"; 
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by (rtac trans_trancl 1);
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qed "trans_less_than";
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AddIffs [trans_less_than];
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goalw thy [less_than_def, less_def] "((x,y): less_than) = (x<y)"; 
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by (Simp_tac 1);
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qed "less_than_iff";
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AddIffs [less_than_iff];
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(*----------------------------------------------------------------------------
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 * The inverse image into a wellfounded relation is wellfounded.
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 *---------------------------------------------------------------------------*)
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goal thy "!!r. wf(r) ==> wf(inv_image r (f::'a=>'b))"; 
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by (full_simp_tac (!simpset addsimps [inv_image_def, wf_eq_minimal]) 1);
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by (Step_tac 1);
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by (subgoal_tac "? (w::'b). w : {w. ? (x::'a). x: Q & (f x = w)}" 1);
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by (blast_tac (!claset delrules [allE]) 2);
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by (etac allE 1);
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by (mp_tac 1);
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by (Blast_tac 1);
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qed "wf_inv_image";
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AddSIs [wf_inv_image];
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goalw thy [trans_def,inv_image_def]
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    "!!r. trans r ==> trans (inv_image r f)";
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by (Simp_tac 1);
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by (Blast_tac 1);
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qed "trans_inv_image";
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(*----------------------------------------------------------------------------
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 * All measures are wellfounded.
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 *---------------------------------------------------------------------------*)
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goalw thy [measure_def] "wf (measure f)";
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by (rtac (wf_less_than RS wf_inv_image) 1);
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qed "wf_measure";
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AddIffs [wf_measure];
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(*----------------------------------------------------------------------------
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 * Wellfoundedness of lexicographic combinations
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 *---------------------------------------------------------------------------*)
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val [wfa,wfb] = goalw thy [wf_def,lex_prod_def]
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 "[| wf(ra); wf(rb) |] ==> wf(ra**rb)";
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by (EVERY1 [rtac allI,rtac impI]);
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by (simp_tac (HOL_basic_ss addsimps [split_paired_All]) 1);
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by (rtac (wfa RS spec RS mp) 1);
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by (EVERY1 [rtac allI,rtac impI]);
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by (rtac (wfb RS spec RS mp) 1);
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by (Blast_tac 1);
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qed "wf_lex_prod";
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AddSIs [wf_lex_prod];
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(*---------------------------------------------------------------------------
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 * Transitivity of WF combinators.
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 *---------------------------------------------------------------------------*)
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goalw thy [trans_def, lex_prod_def]
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    "!!R1 R2. [| trans R1; trans R2 |] ==> trans (R1 ** R2)";
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by (Simp_tac 1);
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by (Blast_tac 1);
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qed "trans_lex_prod";
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AddSIs [trans_lex_prod];
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(*---------------------------------------------------------------------------
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 * Wellfoundedness of proper subset on finite sets.
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 *---------------------------------------------------------------------------*)
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goalw thy [finite_psubset_def] "wf(finite_psubset)";
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by (rtac (wf_measure RS wf_subset) 1);
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by (simp_tac (!simpset addsimps [measure_def, inv_image_def, less_than_def,
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				 symmetric less_def])1);
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by (fast_tac (!claset addIs [psubset_card]) 1);
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qed "wf_finite_psubset";
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goalw thy [finite_psubset_def, trans_def] "trans finite_psubset";
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by (simp_tac (!simpset addsimps [psubset_def]) 1);
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by (Blast_tac 1);
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qed "trans_finite_psubset";
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(*---------------------------------------------------------------------------
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 * Wellfoundedness of finite acyclic relations
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 * Cannot go into WF because it needs Finite
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 *---------------------------------------------------------------------------*)
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goal thy "!!r. finite r ==> acyclic r --> wf r";
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by (etac finite_induct 1);
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 by (Blast_tac 1);
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by (split_all_tac 1);
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by (Asm_full_simp_tac 1);
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qed_spec_mp "finite_acyclic_wf";
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goal thy "!!r. finite r ==> wf r = acyclic r";
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by (blast_tac (!claset addIs [finite_acyclic_wf,wf_acyclic]) 1);
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qed "wf_iff_acyclic_if_finite";
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(*---------------------------------------------------------------------------
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 * A relation is wellfounded iff it has no infinite descending chain
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 *---------------------------------------------------------------------------*)
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goalw thy [wf_eq_minimal RS eq_reflection]
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  "wf r = (~(? f. !i. (f(Suc i),f i) : r))";
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by (rtac iffI 1);
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 by (rtac notI 1);
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 by (etac exE 1);
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 by (eres_inst_tac [("x","{w. ? i. w=f i}")] allE 1);
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 by (Blast_tac 1);
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by (etac swap 1);
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by (Asm_full_simp_tac 1);
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by (Step_tac 1);
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by (subgoal_tac "!n. nat_rec x (%i y. @z. z:Q & (z,y):r) n : Q" 1);
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 by (res_inst_tac[("x","nat_rec x (%i y. @z. z:Q & (z,y):r)")]exI 1);
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 by (rtac allI 1);
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 by (Simp_tac 1);
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 by (rtac selectI2EX 1);
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  by (Blast_tac 1);
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 by (Blast_tac 1);
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by (rtac allI 1);
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by (induct_tac "n" 1);
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 by (Asm_simp_tac 1);
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by (Simp_tac 1);
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by (rtac selectI2EX 1);
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 by (Blast_tac 1);
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by (Blast_tac 1);
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qed "wf_iff_no_infinite_down_chain";