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(* Title: HOL/ex/Perm.ML
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1995 University of Cambridge
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Permutations: example of an inductive definition
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*)
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(*It would be nice to prove
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xs <~~> ys = (!x. count xs x = count ys x)
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See mset on HOL/ex/Sorting.thy
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*)
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open Perm;
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goal Perm.thy "l <~~> l";
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by (list.induct_tac "l" 1);
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by (REPEAT (ares_tac perm.intrs 1));
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qed "perm_refl";
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val perm_induct = standard (perm.mutual_induct RS spec RS spec RSN (2,rev_mp));
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1192
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(** Some examples of rule induction on permutations **)
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(*The form of the premise lets the induction bind xs and ys.*)
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goal Perm.thy "!!xs. xs <~~> ys ==> xs=[] --> ys=[]";
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by (etac perm_induct 1);
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by (ALLGOALS (asm_simp_tac (HOL_ss addsimps list.simps)));
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qed "perm_Nil_lemma";
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(*A more general version is actually easier to understand!*)
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goal Perm.thy "!!xs. xs <~~> ys ==> length(xs) = length(ys)";
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by (etac perm_induct 1);
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by (ALLGOALS (asm_simp_tac list_ss));
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qed "perm_length";
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goal Perm.thy "!!xs. xs <~~> ys ==> ys <~~> xs";
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by (etac perm_induct 1);
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by (REPEAT (ares_tac perm.intrs 1));
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qed "perm_sym";
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goal Perm.thy "!!xs. [| xs <~~> ys |] ==> x mem xs --> x mem ys";
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by (etac perm_induct 1);
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by (fast_tac HOL_cs 4);
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by (ALLGOALS (asm_simp_tac (list_ss setloop split_tac [expand_if])));
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val perm_mem_lemma = result();
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bind_thm ("perm_mem", perm_mem_lemma RS mp);
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(** Ways of making new permutations **)
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(*We can insert the head anywhere in the list*)
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goal Perm.thy "a # xs @ ys <~~> xs @ a # ys";
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by (list.induct_tac "xs" 1);
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by (simp_tac (list_ss addsimps [perm_refl]) 1);
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by (simp_tac list_ss 1);
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by (etac ([perm.swap, perm.Cons] MRS perm.trans) 1);
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qed "perm_append_Cons";
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(*single steps
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by (rtac perm.trans 1);
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by (rtac perm.swap 1);
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by (rtac perm.Cons 1);
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*)
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goal Perm.thy "xs@ys <~~> ys@xs";
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by (list.induct_tac "xs" 1);
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by (simp_tac (list_ss addsimps [perm_refl]) 1);
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by (simp_tac list_ss 1);
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by (etac ([perm.Cons, perm_append_Cons] MRS perm.trans) 1);
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qed "perm_append_swap";
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goal Perm.thy "a # xs <~~> xs @ [a]";
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by (rtac perm.trans 1);
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br perm_append_swap 2;
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by (simp_tac (list_ss addsimps [perm_refl]) 1);
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qed "perm_append_single";
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goal Perm.thy "rev xs <~~> xs";
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by (list.induct_tac "xs" 1);
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by (simp_tac (list_ss addsimps [perm_refl]) 1);
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by (simp_tac list_ss 1);
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by (rtac (perm_append_single RS perm_sym RS perm.trans) 1);
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by (etac perm.Cons 1);
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qed "perm_rev";
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goal Perm.thy "!!xs. xs <~~> ys ==> l@xs <~~> l@ys";
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by (list.induct_tac "l" 1);
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by (simp_tac list_ss 1);
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by (asm_simp_tac (list_ss addsimps [perm.Cons]) 1);
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qed "perm_append1";
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goal Perm.thy "!!xs. xs <~~> ys ==> xs@l <~~> ys@l";
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by (rtac (perm_append_swap RS perm.trans) 1);
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by (etac (perm_append1 RS perm.trans) 1);
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by (rtac perm_append_swap 1);
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qed "perm_append2";
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