author  oheimb 
Tue, 24 Mar 1998 15:49:32 +0100  
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parent 4749  35b47e36e615 
child 5069  3ea049f7979d 
permissions  rwrr 
3193  1 
(* 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 (Clarify_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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val measure_induct = standard 
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(asm_full_simplify (simpset() addsimps [measure_def,inv_image_def]) 

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(wf_measure RS wf_induct)); 

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store_thm("measure_induct",measure_induct); 

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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() addSIs [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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qed_goal "finite_acyclic_wf_converse" thy 
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"!!X. [finite r; acyclic r] ==> wf (r^1)" (K [ 
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etac (finite_converse RS iffD2 RS finite_acyclic_wf) 1, 
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etac (acyclic_converse RS iffD2) 1]); 

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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 (Clarify_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"; 