src/HOL/Library/Permutation.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Permutation.thy	Sun Feb 04 19:42:54 2001 +0100
@@ -0,0 +1,199 @@
+(*  Title:      HOL/Library/Permutation.thy
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1995  University of Cambridge
+
+TODO: it would be nice to prove (for "multiset", defined on
+HOL/ex/Sorting.thy) xs <~~> ys = (\<forall>x. multiset xs x = multiset ys x)
+*)
+
+header {*
+ \title{Permutations}
+ \author{Lawrence C Paulson and Thomas M Rasmussen}
+*}
+
+theory Permutation = Main:
+
+consts
+  perm :: "('a list * 'a list) set"
+
+syntax
+  "_perm" :: "'a list => 'a list => bool"    ("_ <~~> _"  [50, 50] 50)
+translations
+  "x <~~> y" == "(x, y) \<in> perm"
+
+inductive perm
+  intros [intro]
+    Nil: "[] <~~> []"
+    swap: "y # x # l <~~> x # y # l"
+    Cons: "xs <~~> ys ==> z # xs <~~> z # ys"
+    trans: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
+
+lemma perm_refl [iff]: "l <~~> l"
+  apply (induct l)
+   apply auto
+  done
+
+
+subsection {* Some examples of rule induction on permutations *}
+
+lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
+    -- {* the form of the premise lets the induction bind @{term xs} and @{term ys} *}
+  apply (erule perm.induct)
+     apply (simp_all (no_asm_simp))
+  done
+
+lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
+  apply (insert xperm_empty_imp_aux)
+  apply blast
+  done
+
+
+text {*
+  \medskip This more general theorem is easier to understand!
+  *}
+
+lemma perm_length: "xs <~~> ys ==> length xs = length ys"
+  apply (erule perm.induct)
+     apply simp_all
+  done
+
+lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
+  apply (drule perm_length)
+  apply auto
+  done
+
+lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
+  apply (erule perm.induct)
+     apply auto
+  done
+
+lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
+  apply (erule perm.induct)
+     apply auto
+  done
+
+
+subsection {* Ways of making new permutations *}
+
+text {*
+  We can insert the head anywhere in the list.
+*}
+
+lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
+  apply (induct xs)
+   apply auto
+  done
+
+lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
+  apply (induct xs)
+    apply simp_all
+  apply (blast intro: perm_append_Cons)
+  done
+
+lemma perm_append_single: "a # xs <~~> xs @ [a]"
+  apply (rule perm.trans)
+   prefer 2
+   apply (rule perm_append_swap)
+  apply simp
+  done
+
+lemma perm_rev: "rev xs <~~> xs"
+  apply (induct xs)
+   apply simp_all
+  apply (blast intro: perm_sym perm_append_single)
+  done
+
+lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
+  apply (induct l)
+   apply auto
+  done
+
+lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
+  apply (blast intro!: perm_append_swap perm_append1)
+  done
+
+
+subsection {* Further results *}
+
+lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
+  apply (blast intro: perm_empty_imp)
+  done
+
+lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
+  apply auto
+  apply (erule perm_sym [THEN perm_empty_imp])
+  done
+
+lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
+  apply (erule perm.induct)
+     apply auto
+  done
+
+lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
+  apply (blast intro: perm_sing_imp)
+  done
+
+lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
+  apply (blast dest: perm_sym)
+  done
+
+
+subsection {* Removing elements *}
+
+consts
+  remove :: "'a => 'a list => 'a list"
+primrec
+  "remove x [] = []"
+  "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
+
+lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
+  apply (induct ys)
+   apply auto
+  done
+
+lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
+  apply (induct l)
+   apply auto
+  done
+
+
+text {* \medskip Congruence rule *}
+
+lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
+  apply (erule perm.induct)
+     apply auto
+  done
+
+lemma remove_hd [simp]: "remove z (z # xs) = xs"
+  apply auto
+  done
+
+lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
+  apply (drule_tac z = z in perm_remove_perm)
+  apply auto
+  done
+
+lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
+  apply (blast intro: cons_perm_imp_perm)
+  done
+
+lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
+  apply (induct zs rule: rev_induct)
+   apply (simp_all (no_asm_use))
+  apply blast
+  done
+
+lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
+  apply (blast intro: append_perm_imp_perm perm_append1)
+  done
+
+lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
+  apply (safe intro!: perm_append2)
+  apply (rule append_perm_imp_perm)
+  apply (rule perm_append_swap [THEN perm.trans])
+    -- {* the previous step helps this @{text blast} call succeed quickly *}
+  apply (blast intro: perm_append_swap)
+  done
+
+end