author | paulson |
Fri, 29 Oct 2004 15:16:02 +0200 | |
changeset 15270 | 8b3f707a78a7 |
parent 13867 | 1fdecd15437f |
permissions | -rw-r--r-- |
10213 | 1 |
(* Title: HOL/Wellfounded_Relations |
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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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section "`Less than' on the natural numbers"; |
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Goalw [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 [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 [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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Goal "(!!n. (ALL m. Suc m <= n --> P m) ==> P n) ==> P n"; |
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by (rtac (wf_less_than RS wf_induct) 1); |
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by (resolve_tac (premises()) 1); |
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by Auto_tac; |
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qed_spec_mp "full_nat_induct"; |
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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 "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 "EX (w::'b). w : {w. EX (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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Addsimps [wf_inv_image]; |
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AddSIs [wf_inv_image]; |
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(*---------------------------------------------------------------------------- |
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* All measures are wellfounded. |
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*---------------------------------------------------------------------------*) |
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Goalw [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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bind_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 (the_context ()) [wf_def,lex_prod_def] |
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"[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> 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 [trans_def, lex_prod_def] |
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"!!R1 R2. [| trans R1; trans R2 |] ==> trans (R1 <*lex*> 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 [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() addSEs [psubset_card_mono]) 1); |
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qed "wf_finite_psubset"; |
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Goalw [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 "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 "[|finite r; acyclic r|] ==> wf (r^-1)"; |
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by (etac (finite_converse RS iffD2 RS finite_acyclic_wf) 1); |
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by (etac (acyclic_converse RS iffD2) 1); |
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qed "finite_acyclic_wf_converse"; |
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Goal "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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* Weakly decreasing sequences (w.r.t. some well-founded order) stabilize. |
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*---------------------------------------------------------------------------*) |
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Goal "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"; |
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by (induct_tac "k" 1); |
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by (ALLGOALS Simp_tac); |
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by (blast_tac (claset() addIs [rtrancl_trans]) 1); |
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val lemma = result(); |
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Goal "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |] \ |
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\ ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"; |
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by (etac wf_induct 1); |
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by (Clarify_tac 1); |
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by (case_tac "EX j. (f (m+j), f m) : r^+" 1); |
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by (Clarify_tac 1); |
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by (subgoal_tac "EX i. ALL k. f ((m+j)+i+k) = f ((m+j)+i)" 1); |
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by (Clarify_tac 1); |
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by (res_inst_tac [("x","j+i")] exI 1); |
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by (asm_full_simp_tac (simpset() addsimps add_ac) 1); |
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by (Blast_tac 1); |
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by (res_inst_tac [("x","0")] exI 1); |
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by (Clarsimp_tac 1); |
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by (dres_inst_tac [("i","m"), ("k","k")] lemma 1); |
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by (blast_tac (claset() addEs [rtranclE] addDs [rtrancl_into_trancl1]) 1); |
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val lemma = result(); |
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Goal "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |] \ |
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\ ==> EX i. ALL k. f (i+k) = f i"; |
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by (dres_inst_tac [("x","0")] (lemma RS spec) 1); |
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by Auto_tac; |
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qed "wf_weak_decr_stable"; |
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11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11340
diff
changeset
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(* special case of the theorem above: <= *) |
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Goal "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"; |
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by (res_inst_tac [("r","pred_nat")] wf_weak_decr_stable 1); |
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11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11340
diff
changeset
|
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by (asm_simp_tac (simpset() addsimps [thm "pred_nat_trancl_eq_le"]) 1); |
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by (REPEAT (resolve_tac [wf_trancl,wf_pred_nat] 1)); |
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qed "weak_decr_stable"; |
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(*---------------------------------------------------------------------------- |
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* Wellfoundedness of same_fst |
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*---------------------------------------------------------------------------*) |
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Goalw[same_fst_def] "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"; |
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by (Asm_simp_tac 1); |
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qed "same_fstI"; |
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AddSIs[same_fstI]; |
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val prems = goalw thy [same_fst_def] |
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"(!!x. P x ==> wf(R x)) ==> wf(same_fst P R)"; |
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by (full_simp_tac (simpset() delcongs [imp_cong] addsimps [wf_def]) 1); |
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by (strip_tac 1); |
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by (rename_tac "a b" 1); |
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by (case_tac "wf(R a)" 1); |
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by (eres_inst_tac [("a","b")] wf_induct 1); |
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by (Blast_tac 1); |
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by (blast_tac (claset() addIs prems) 1); |
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qed "wf_same_fst"; |