author | wenzelm |
Mon, 30 Mar 2009 22:38:50 +0200 | |
changeset 30804 | dbdb74be8dde |
parent 30663 | 0b6aff7451b2 |
child 32456 | 341c83339aeb |
permissions | -rw-r--r-- |
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Permutations, both general and specifically on finite sets.
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(* Title: Library/Permutations |
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Permutations, both general and specifically on finite sets.
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Author: Amine Chaieb, University of Cambridge |
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Permutations, both general and specifically on finite sets.
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*) |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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header {* Permutations, both general and specifically on finite sets.*} |
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Permutations, both general and specifically on finite sets.
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|
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Permutations, both general and specifically on finite sets.
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theory Permutations |
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Main is (Complex_Main) base entry point in library theories
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imports Finite_Cartesian_Product Parity Fact Main |
29840
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Permutations, both general and specifically on finite sets.
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begin |
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Permutations, both general and specifically on finite sets.
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|
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Permutations, both general and specifically on finite sets.
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(* Why should I import Main just to solve the Typerep problem! *) |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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definition permutes (infixr "permutes" 41) where |
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Permutations, both general and specifically on finite sets.
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"(p permutes S) \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)" |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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(* ------------------------------------------------------------------------- *) |
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Permutations, both general and specifically on finite sets.
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(* Transpositions. *) |
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Permutations, both general and specifically on finite sets.
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(* ------------------------------------------------------------------------- *) |
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Permutations, both general and specifically on finite sets.
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|
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Permutations, both general and specifically on finite sets.
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declare swap_self[simp] |
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lemma swapid_sym: "Fun.swap a b id = Fun.swap b a id" |
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Permutations, both general and specifically on finite sets.
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by (auto simp add: expand_fun_eq swap_def fun_upd_def) |
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Permutations, both general and specifically on finite sets.
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lemma swap_id_refl: "Fun.swap a a id = id" by simp |
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Permutations, both general and specifically on finite sets.
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lemma swap_id_sym: "Fun.swap a b id = Fun.swap b a id" |
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Permutations, both general and specifically on finite sets.
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by (rule ext, simp add: swap_def) |
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Permutations, both general and specifically on finite sets.
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lemma swap_id_idempotent[simp]: "Fun.swap a b id o Fun.swap a b id = id" |
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Permutations, both general and specifically on finite sets.
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by (rule ext, auto simp add: swap_def) |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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lemma inv_unique_comp: assumes fg: "f o g = id" and gf: "g o f = id" |
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Permutations, both general and specifically on finite sets.
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shows "inv f = g" |
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Permutations, both general and specifically on finite sets.
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using fg gf inv_equality[of g f] by (auto simp add: expand_fun_eq) |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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lemma inverse_swap_id: "inv (Fun.swap a b id) = Fun.swap a b id" |
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Permutations, both general and specifically on finite sets.
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by (rule inv_unique_comp, simp_all) |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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lemma swap_id_eq: "Fun.swap a b id x = (if x = a then b else if x = b then a else x)" |
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Permutations, both general and specifically on finite sets.
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by (simp add: swap_def) |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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(* ------------------------------------------------------------------------- *) |
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Permutations, both general and specifically on finite sets.
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(* Basic consequences of the definition. *) |
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Permutations, both general and specifically on finite sets.
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(* ------------------------------------------------------------------------- *) |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S" |
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Permutations, both general and specifically on finite sets.
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unfolding permutes_def by metis |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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lemma permutes_image: assumes pS: "p permutes S" shows "p ` S = S" |
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Permutations, both general and specifically on finite sets.
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using pS |
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unfolding permutes_def |
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apply - |
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apply (rule set_ext) |
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Permutations, both general and specifically on finite sets.
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apply (simp add: image_iff) |
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Permutations, both general and specifically on finite sets.
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apply metis |
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Permutations, both general and specifically on finite sets.
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done |
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Permutations, both general and specifically on finite sets.
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lemma permutes_inj: "p permutes S ==> inj p " |
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unfolding permutes_def inj_on_def by blast |
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lemma permutes_surj: "p permutes s ==> surj p" |
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unfolding permutes_def surj_def by metis |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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lemma permutes_inv_o: assumes pS: "p permutes S" |
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Permutations, both general and specifically on finite sets.
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shows " p o inv p = id" |
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Permutations, both general and specifically on finite sets.
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and "inv p o p = id" |
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Permutations, both general and specifically on finite sets.
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using permutes_inj[OF pS] permutes_surj[OF pS] |
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Permutations, both general and specifically on finite sets.
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unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+ |
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Permutations, both general and specifically on finite sets.
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|
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Permutations, both general and specifically on finite sets.
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|
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lemma permutes_inverses: |
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Permutations, both general and specifically on finite sets.
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fixes p :: "'a \<Rightarrow> 'a" |
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Permutations, both general and specifically on finite sets.
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assumes pS: "p permutes S" |
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Permutations, both general and specifically on finite sets.
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shows "p (inv p x) = x" |
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Permutations, both general and specifically on finite sets.
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and "inv p (p x) = x" |
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Permutations, both general and specifically on finite sets.
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parents:
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using permutes_inv_o[OF pS, unfolded expand_fun_eq o_def] by auto |
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Permutations, both general and specifically on finite sets.
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|
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Permutations, both general and specifically on finite sets.
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lemma permutes_subset: "p permutes S \<Longrightarrow> S \<subseteq> T ==> p permutes T" |
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Permutations, both general and specifically on finite sets.
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unfolding permutes_def by blast |
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Permutations, both general and specifically on finite sets.
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77 |
|
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Permutations, both general and specifically on finite sets.
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lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id" |
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unfolding expand_fun_eq permutes_def apply simp by metis |
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Permutations, both general and specifically on finite sets.
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|
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Permutations, both general and specifically on finite sets.
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lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id" |
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Permutations, both general and specifically on finite sets.
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parents:
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82 |
unfolding expand_fun_eq permutes_def apply simp by metis |
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Permutations, both general and specifically on finite sets.
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84 |
lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)" |
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Permutations, both general and specifically on finite sets.
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parents:
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85 |
unfolding permutes_def by simp |
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Permutations, both general and specifically on finite sets.
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86 |
|
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Permutations, both general and specifically on finite sets.
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87 |
lemma permutes_inv_eq: "p permutes S ==> inv p y = x \<longleftrightarrow> p x = y" |
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Permutations, both general and specifically on finite sets.
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parents:
diff
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88 |
unfolding permutes_def inv_def apply auto |
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Permutations, both general and specifically on finite sets.
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parents:
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89 |
apply (erule allE[where x=y]) |
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Permutations, both general and specifically on finite sets.
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parents:
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changeset
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90 |
apply (erule allE[where x=y]) |
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Permutations, both general and specifically on finite sets.
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parents:
diff
changeset
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91 |
apply (rule someI_ex) apply blast |
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Permutations, both general and specifically on finite sets.
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parents:
diff
changeset
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92 |
apply (rule some1_equality) |
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Permutations, both general and specifically on finite sets.
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parents:
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|
93 |
apply blast |
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Permutations, both general and specifically on finite sets.
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parents:
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94 |
apply blast |
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Permutations, both general and specifically on finite sets.
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parents:
diff
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95 |
done |
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Permutations, both general and specifically on finite sets.
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parents:
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96 |
|
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Permutations, both general and specifically on finite sets.
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parents:
diff
changeset
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97 |
lemma permutes_swap_id: "a \<in> S \<Longrightarrow> b \<in> S ==> Fun.swap a b id permutes S" |
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Permutations, both general and specifically on finite sets.
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parents:
diff
changeset
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98 |
unfolding permutes_def swap_def fun_upd_def apply auto apply metis done |
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Permutations, both general and specifically on finite sets.
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parents:
diff
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99 |
|
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
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100 |
lemma permutes_superset: "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T" |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
101 |
apply (simp add: Ball_def permutes_def Diff_iff) by metis |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
102 |
|
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Permutations, both general and specifically on finite sets.
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parents:
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|
103 |
(* ------------------------------------------------------------------------- *) |
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Permutations, both general and specifically on finite sets.
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parents:
diff
changeset
|
104 |
(* Group properties. *) |
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Permutations, both general and specifically on finite sets.
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parents:
diff
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|
105 |
(* ------------------------------------------------------------------------- *) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
106 |
|
30488 | 107 |
lemma permutes_id: "id permutes S" unfolding permutes_def by simp |
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Permutations, both general and specifically on finite sets.
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parents:
diff
changeset
|
108 |
|
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Permutations, both general and specifically on finite sets.
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parents:
diff
changeset
|
109 |
lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S ==> q o p permutes S" |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
110 |
unfolding permutes_def o_def by metis |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
111 |
|
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
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112 |
lemma permutes_inv: assumes pS: "p permutes S" shows "inv p permutes S" |
30488 | 113 |
using pS unfolding permutes_def permutes_inv_eq[OF pS] by metis |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
114 |
|
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Permutations, both general and specifically on finite sets.
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parents:
diff
changeset
|
115 |
lemma permutes_inv_inv: assumes pS: "p permutes S" shows "inv (inv p) = p" |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
116 |
unfolding expand_fun_eq permutes_inv_eq[OF pS] permutes_inv_eq[OF permutes_inv[OF pS]] |
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Permutations, both general and specifically on finite sets.
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parents:
diff
changeset
|
117 |
by blast |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
118 |
|
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
119 |
(* ------------------------------------------------------------------------- *) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
120 |
(* The number of permutations on a finite set. *) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
121 |
(* ------------------------------------------------------------------------- *) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
122 |
|
30488 | 123 |
lemma permutes_insert_lemma: |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
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124 |
assumes pS: "p permutes (insert a S)" |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
125 |
shows "Fun.swap a (p a) id o p permutes S" |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
126 |
apply (rule permutes_superset[where S = "insert a S"]) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
127 |
apply (rule permutes_compose[OF pS]) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
128 |
apply (rule permutes_swap_id, simp) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
129 |
using permutes_in_image[OF pS, of a] apply simp |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
130 |
apply (auto simp add: Ball_def Diff_iff swap_def) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
131 |
done |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
132 |
|
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
133 |
lemma permutes_insert: "{p. p permutes (insert a S)} = |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
134 |
(\<lambda>(b,p). Fun.swap a b id o p) ` {(b,p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}" |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
135 |
proof- |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
136 |
|
30488 | 137 |
{fix p |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
138 |
{assume pS: "p permutes insert a S" |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
139 |
let ?b = "p a" |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
140 |
let ?q = "Fun.swap a (p a) id o p" |
30488 | 141 |
have th0: "p = Fun.swap a ?b id o ?q" unfolding expand_fun_eq o_assoc by simp |
142 |
have th1: "?b \<in> insert a S " unfolding permutes_in_image[OF pS] by simp |
|
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
143 |
from permutes_insert_lemma[OF pS] th0 th1 |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
144 |
have "\<exists> b q. p = Fun.swap a b id o q \<and> b \<in> insert a S \<and> q permutes S" by blast} |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
145 |
moreover |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
146 |
{fix b q assume bq: "p = Fun.swap a b id o q" "b \<in> insert a S" "q permutes S" |
30488 | 147 |
from permutes_subset[OF bq(3), of "insert a S"] |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
148 |
have qS: "q permutes insert a S" by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
149 |
have aS: "a \<in> insert a S" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
150 |
from bq(1) permutes_compose[OF qS permutes_swap_id[OF aS bq(2)]] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
151 |
have "p permutes insert a S" by simp } |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
152 |
ultimately have "p permutes insert a S \<longleftrightarrow> (\<exists> b q. p = Fun.swap a b id o q \<and> b \<in> insert a S \<and> q permutes S)" by blast} |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
153 |
thus ?thesis by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
154 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
155 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
156 |
lemma hassize_insert: "a \<notin> F \<Longrightarrow> insert a F hassize n \<Longrightarrow> F hassize (n - 1)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
157 |
by (auto simp add: hassize_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
158 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
159 |
lemma hassize_permutations: assumes Sn: "S hassize n" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
160 |
shows "{p. p permutes S} hassize (fact n)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
161 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
162 |
from Sn have fS:"finite S" by (simp add: hassize_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
163 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
164 |
have "\<forall>n. (S hassize n) \<longrightarrow> ({p. p permutes S} hassize (fact n))" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
165 |
proof(rule finite_induct[where F = S]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
166 |
from fS show "finite S" . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
167 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
168 |
show "\<forall>n. ({} hassize n) \<longrightarrow> ({p. p permutes {}} hassize fact n)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
169 |
by (simp add: hassize_def permutes_empty) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
170 |
next |
30488 | 171 |
fix x F |
172 |
assume fF: "finite F" and xF: "x \<notin> F" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
173 |
and H: "\<forall>n. (F hassize n) \<longrightarrow> ({p. p permutes F} hassize fact n)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
174 |
{fix n assume H0: "insert x F hassize n" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
175 |
let ?xF = "{p. p permutes insert x F}" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
176 |
let ?pF = "{p. p permutes F}" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
177 |
let ?pF' = "{(b, p). b \<in> insert x F \<and> p \<in> ?pF}" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
178 |
let ?g = "(\<lambda>(b, p). Fun.swap x b id \<circ> p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
179 |
from permutes_insert[of x F] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
180 |
have xfgpF': "?xF = ?g ` ?pF'" . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
181 |
from hassize_insert[OF xF H0] have Fs: "F hassize (n - 1)" . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
182 |
from H Fs have pFs: "?pF hassize fact (n - 1)" by blast |
30488 | 183 |
hence pF'f: "finite ?pF'" using H0 unfolding hassize_def |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
184 |
apply (simp only: Collect_split Collect_mem_eq) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
185 |
apply (rule finite_cartesian_product) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
186 |
apply simp_all |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
187 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
188 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
189 |
have ginj: "inj_on ?g ?pF'" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
190 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
191 |
{ |
30488 | 192 |
fix b p c q assume bp: "(b,p) \<in> ?pF'" and cq: "(c,q) \<in> ?pF'" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
193 |
and eq: "?g (b,p) = ?g (c,q)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
194 |
from bp cq have ths: "b \<in> insert x F" "c \<in> insert x F" "x \<in> insert x F" "p permutes F" "q permutes F" by auto |
30488 | 195 |
from ths(4) xF eq have "b = ?g (b,p) x" unfolding permutes_def |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
196 |
by (auto simp add: swap_def fun_upd_def expand_fun_eq) |
30488 | 197 |
also have "\<dots> = ?g (c,q) x" using ths(5) xF eq |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
198 |
by (auto simp add: swap_def fun_upd_def expand_fun_eq) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
199 |
also have "\<dots> = c"using ths(5) xF unfolding permutes_def |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
200 |
by (auto simp add: swap_def fun_upd_def expand_fun_eq) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
201 |
finally have bc: "b = c" . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
202 |
hence "Fun.swap x b id = Fun.swap x c id" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
203 |
with eq have "Fun.swap x b id o p = Fun.swap x b id o q" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
204 |
hence "Fun.swap x b id o (Fun.swap x b id o p) = Fun.swap x b id o (Fun.swap x b id o q)" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
205 |
hence "p = q" by (simp add: o_assoc) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
206 |
with bc have "(b,p) = (c,q)" by simp } |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
207 |
thus ?thesis unfolding inj_on_def by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
208 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
209 |
from xF H0 have n0: "n \<noteq> 0 " by (auto simp add: hassize_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
210 |
hence "\<exists>m. n = Suc m" by arith |
30488 | 211 |
then obtain m where n[simp]: "n = Suc m" by blast |
212 |
from pFs H0 have xFc: "card ?xF = fact n" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
213 |
unfolding xfgpF' card_image[OF ginj] hassize_def |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
214 |
apply (simp only: Collect_split Collect_mem_eq card_cartesian_product) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
215 |
by simp |
30488 | 216 |
from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF" unfolding xfgpF' by simp |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
217 |
have "?xF hassize fact n" |
30488 | 218 |
using xFf xFc |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
219 |
unfolding hassize_def xFf by blast } |
30488 | 220 |
thus "\<forall>n. (insert x F hassize n) \<longrightarrow> ({p. p permutes insert x F} hassize fact n)" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
221 |
by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
222 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
223 |
with Sn show ?thesis by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
224 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
225 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
226 |
lemma finite_permutations: "finite S ==> finite {p. p permutes S}" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
227 |
using hassize_permutations[of S] unfolding hassize_def by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
228 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
229 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
230 |
(* Permutations of index set for iterated operations. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
231 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
232 |
|
30488 | 233 |
lemma (in ab_semigroup_mult) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
234 |
shows "fold_image times f z S = fold_image times (f o p) z S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
235 |
using fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
236 |
unfolding permutes_image[OF pS] . |
30488 | 237 |
lemma (in ab_semigroup_add) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
238 |
shows "fold_image plus f z S = fold_image plus (f o p) z S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
239 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
240 |
interpret ab_semigroup_mult plus apply unfold_locales apply (simp add: add_assoc) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
241 |
apply (simp add: add_commute) done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
242 |
from fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
243 |
show ?thesis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
244 |
unfolding permutes_image[OF pS] . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
245 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
246 |
|
30488 | 247 |
lemma setsum_permute: assumes pS: "p permutes S" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
248 |
shows "setsum f S = setsum (f o p) S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
249 |
unfolding setsum_def using fold_image_permute[of S p f 0] pS by clarsimp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
250 |
|
30488 | 251 |
lemma setsum_permute_natseg:assumes pS: "p permutes {m .. n}" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
252 |
shows "setsum f {m .. n} = setsum (f o p) {m .. n}" |
30488 | 253 |
using setsum_permute[OF pS, of f ] pS by blast |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
254 |
|
30488 | 255 |
lemma setprod_permute: assumes pS: "p permutes S" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
256 |
shows "setprod f S = setprod (f o p) S" |
30488 | 257 |
unfolding setprod_def |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
258 |
using ab_semigroup_mult_class.fold_image_permute[of S p f 1] pS by clarsimp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
259 |
|
30488 | 260 |
lemma setprod_permute_natseg:assumes pS: "p permutes {m .. n}" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
261 |
shows "setprod f {m .. n} = setprod (f o p) {m .. n}" |
30488 | 262 |
using setprod_permute[OF pS, of f ] pS by blast |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
263 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
264 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
265 |
(* Various combinations of transpositions with 2, 1 and 0 common elements. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
266 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
267 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
268 |
lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow> Fun.swap a b id o Fun.swap a c id = Fun.swap b c id o Fun.swap a b id" by (simp add: expand_fun_eq swap_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
269 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
270 |
lemma swap_id_common': "~(a = b) \<Longrightarrow> ~(a = c) \<Longrightarrow> Fun.swap a c id o Fun.swap b c id = Fun.swap b c id o Fun.swap a b id" by (simp add: expand_fun_eq swap_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
271 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
272 |
lemma swap_id_independent: "~(a = c) \<Longrightarrow> ~(a = d) \<Longrightarrow> ~(b = c) \<Longrightarrow> ~(b = d) ==> Fun.swap a b id o Fun.swap c d id = Fun.swap c d id o Fun.swap a b id" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
273 |
by (simp add: swap_def expand_fun_eq) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
274 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
275 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
276 |
(* Permutations as transposition sequences. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
277 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
278 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
279 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
280 |
inductive swapidseq :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" where |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
281 |
id[simp]: "swapidseq 0 id" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
282 |
| comp_Suc: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (Fun.swap a b id o p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
283 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
284 |
declare id[unfolded id_def, simp] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
285 |
definition "permutation p \<longleftrightarrow> (\<exists>n. swapidseq n p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
286 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
287 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
288 |
(* Some closure properties of the set of permutations, with lengths. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
289 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
290 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
291 |
lemma permutation_id[simp]: "permutation id"unfolding permutation_def |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
292 |
by (rule exI[where x=0], simp) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
293 |
declare permutation_id[unfolded id_def, simp] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
294 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
295 |
lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
296 |
apply clarsimp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
297 |
using comp_Suc[of 0 id a b] by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
298 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
299 |
lemma permutation_swap_id: "permutation (Fun.swap a b id)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
300 |
apply (cases "a=b", simp_all) |
30488 | 301 |
unfolding permutation_def using swapidseq_swap[of a b] by blast |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
302 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
303 |
lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q ==> swapidseq (n + m) (p o q)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
304 |
proof (induct n p arbitrary: m q rule: swapidseq.induct) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
305 |
case (id m q) thus ?case by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
306 |
next |
30488 | 307 |
case (comp_Suc n p a b m q) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
308 |
have th: "Suc n + m = Suc (n + m)" by arith |
30488 | 309 |
show ?case unfolding th o_assoc[symmetric] |
310 |
apply (rule swapidseq.comp_Suc) using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3) by blast+ |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
311 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
312 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
313 |
lemma permutation_compose: "permutation p \<Longrightarrow> permutation q ==> permutation(p o q)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
314 |
unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
315 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
316 |
lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b ==> swapidseq (Suc n) (p o Fun.swap a b id)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
317 |
apply (induct n p rule: swapidseq.induct) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
318 |
using swapidseq_swap[of a b] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
319 |
by (auto simp add: o_assoc[symmetric] intro: swapidseq.comp_Suc) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
320 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
321 |
lemma swapidseq_inverse_exists: "swapidseq n p ==> \<exists>q. swapidseq n q \<and> p o q = id \<and> q o p = id" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
322 |
proof(induct n p rule: swapidseq.induct) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
323 |
case id thus ?case by (rule exI[where x=id], simp) |
30488 | 324 |
next |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
325 |
case (comp_Suc n p a b) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
326 |
from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
327 |
let ?q = "q o Fun.swap a b id" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
328 |
note H = comp_Suc.hyps |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
329 |
from swapidseq_swap[of a b] H(3) have th0: "swapidseq 1 (Fun.swap a b id)" by simp |
30488 | 330 |
from swapidseq_comp_add[OF q(1) th0] have th1:"swapidseq (Suc n) ?q" by simp |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
331 |
have "Fun.swap a b id o p o ?q = Fun.swap a b id o (p o q) o Fun.swap a b id" by (simp add: o_assoc) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
332 |
also have "\<dots> = id" by (simp add: q(2)) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
333 |
finally have th2: "Fun.swap a b id o p o ?q = id" . |
30488 | 334 |
have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id o Fun.swap a b id) \<circ> p" by (simp only: o_assoc) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
335 |
hence "?q \<circ> (Fun.swap a b id \<circ> p) = id" by (simp add: q(3)) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
336 |
with th1 th2 show ?case by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
337 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
338 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
339 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
340 |
lemma swapidseq_inverse: assumes H: "swapidseq n p" shows "swapidseq n (inv p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
341 |
using swapidseq_inverse_exists[OF H] inv_unique_comp[of p] by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
342 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
343 |
lemma permutation_inverse: "permutation p ==> permutation (inv p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
344 |
using permutation_def swapidseq_inverse by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
345 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
346 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
347 |
(* The identity map only has even transposition sequences. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
348 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
349 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
350 |
lemma symmetry_lemma:"(\<And>a b c d. P a b c d ==> P a b d c) \<Longrightarrow> |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
351 |
(\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> (a = c \<and> b = d \<or> a = c \<and> b \<noteq> d \<or> a \<noteq> c \<and> b = d \<or> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d) ==> P a b c d) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
352 |
==> (\<And>a b c d. a \<noteq> b --> c \<noteq> d \<longrightarrow> P a b c d)" by metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
353 |
|
30488 | 354 |
lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> Fun.swap a b id o Fun.swap c d id = id \<or> |
355 |
(\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id)" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
356 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
357 |
assume H: "a\<noteq>b" "c\<noteq>d" |
30488 | 358 |
have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow> |
359 |
( Fun.swap a b id o Fun.swap c d id = id \<or> |
|
360 |
(\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id))" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
361 |
apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d]) |
30488 | 362 |
apply (simp_all only: swapid_sym) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
363 |
apply (case_tac "a = c \<and> b = d", clarsimp simp only: swapid_sym swap_id_idempotent) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
364 |
apply (case_tac "a = c \<and> b \<noteq> d") |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
365 |
apply (rule disjI2) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
366 |
apply (rule_tac x="b" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
367 |
apply (rule_tac x="d" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
368 |
apply (rule_tac x="b" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
369 |
apply (clarsimp simp add: expand_fun_eq swap_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
370 |
apply (case_tac "a \<noteq> c \<and> b = d") |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
371 |
apply (rule disjI2) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
372 |
apply (rule_tac x="c" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
373 |
apply (rule_tac x="d" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
374 |
apply (rule_tac x="c" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
375 |
apply (clarsimp simp add: expand_fun_eq swap_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
376 |
apply (rule disjI2) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
377 |
apply (rule_tac x="c" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
378 |
apply (rule_tac x="d" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
379 |
apply (rule_tac x="b" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
380 |
apply (clarsimp simp add: expand_fun_eq swap_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
381 |
done |
30488 | 382 |
with H show ?thesis by metis |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
383 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
384 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
385 |
lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
386 |
using swapidseq.cases[of 0 p "p = id"] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
387 |
by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
388 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
389 |
lemma swapidseq_cases: "swapidseq n p \<longleftrightarrow> (n=0 \<and> p = id \<or> (\<exists>a b q m. n = Suc m \<and> p = Fun.swap a b id o q \<and> swapidseq m q \<and> a\<noteq> b))" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
390 |
apply (rule iffI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
391 |
apply (erule swapidseq.cases[of n p]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
392 |
apply simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
393 |
apply (rule disjI2) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
394 |
apply (rule_tac x= "a" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
395 |
apply (rule_tac x= "b" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
396 |
apply (rule_tac x= "pa" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
397 |
apply (rule_tac x= "na" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
398 |
apply simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
399 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
400 |
apply (rule comp_Suc, simp_all) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
401 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
402 |
lemma fixing_swapidseq_decrease: |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
403 |
assumes spn: "swapidseq n p" and ab: "a\<noteq>b" and pa: "(Fun.swap a b id o p) a = a" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
404 |
shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id o p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
405 |
using spn ab pa |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
406 |
proof(induct n arbitrary: p a b) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
407 |
case 0 thus ?case by (auto simp add: swap_def fun_upd_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
408 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
409 |
case (Suc n p a b) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
410 |
from Suc.prems(1) swapidseq_cases[of "Suc n" p] obtain |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
411 |
c d q m where cdqm: "Suc n = Suc m" "p = Fun.swap c d id o q" "swapidseq m q" "c \<noteq> d" "n = m" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
412 |
by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
413 |
{assume H: "Fun.swap a b id o Fun.swap c d id = id" |
30488 | 414 |
|
415 |
have ?case apply (simp only: cdqm o_assoc H) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
416 |
by (simp add: cdqm)} |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
417 |
moreover |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
418 |
{ fix x y z |
30488 | 419 |
assume H: "x\<noteq>a" "y\<noteq>a" "z \<noteq>a" "x \<noteq>y" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
420 |
"Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
421 |
from H have az: "a \<noteq> z" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
422 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
423 |
{fix h have "(Fun.swap x y id o h) a = a \<longleftrightarrow> h a = a" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
424 |
using H by (simp add: swap_def)} |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
425 |
note th3 = this |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
426 |
from cdqm(2) have "Fun.swap a b id o p = Fun.swap a b id o (Fun.swap c d id o q)" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
427 |
hence "Fun.swap a b id o p = Fun.swap x y id o (Fun.swap a z id o q)" by (simp add: o_assoc H) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
428 |
hence "(Fun.swap a b id o p) a = (Fun.swap x y id o (Fun.swap a z id o q)) a" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
429 |
hence "(Fun.swap x y id o (Fun.swap a z id o q)) a = a" unfolding Suc by metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
430 |
hence th1: "(Fun.swap a z id o q) a = a" unfolding th3 . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
431 |
from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az th1] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
432 |
have th2: "swapidseq (n - 1) (Fun.swap a z id o q)" "n \<noteq> 0" by blast+ |
30488 | 433 |
have th: "Suc n - 1 = Suc (n - 1)" using th2(2) by auto |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
434 |
have ?case unfolding cdqm(2) H o_assoc th |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
435 |
apply (simp only: Suc_not_Zero simp_thms o_assoc[symmetric]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
436 |
apply (rule comp_Suc) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
437 |
using th2 H apply blast+ |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
438 |
done} |
30488 | 439 |
ultimately show ?case using swap_general[OF Suc.prems(2) cdqm(4)] by metis |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
440 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
441 |
|
30488 | 442 |
lemma swapidseq_identity_even: |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
443 |
assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)" shows "even n" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
444 |
using `swapidseq n id` |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
445 |
proof(induct n rule: nat_less_induct) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
446 |
fix n |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
447 |
assume H: "\<forall>m<n. swapidseq m (id::'a \<Rightarrow> 'a) \<longrightarrow> even m" "swapidseq n (id :: 'a \<Rightarrow> 'a)" |
30488 | 448 |
{assume "n = 0" hence "even n" by arith} |
449 |
moreover |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
450 |
{fix a b :: 'a and q m |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
451 |
assume h: "n = Suc m" "(id :: 'a \<Rightarrow> 'a) = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
452 |
from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
453 |
have m: "m \<noteq> 0" "swapidseq (m - 1) (id :: 'a \<Rightarrow> 'a)" by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
454 |
from h m have mn: "m - 1 < n" by arith |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
455 |
from H(1)[rule_format, OF mn m(2)] h(1) m(1) have "even n" apply arith done} |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
456 |
ultimately show "even n" using H(2)[unfolded swapidseq_cases[of n id]] by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
457 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
458 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
459 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
460 |
(* Therefore we have a welldefined notion of parity. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
461 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
462 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
463 |
definition "evenperm p = even (SOME n. swapidseq n p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
464 |
|
30488 | 465 |
lemma swapidseq_even_even: assumes |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
466 |
m: "swapidseq m p" and n: "swapidseq n p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
467 |
shows "even m \<longleftrightarrow> even n" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
468 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
469 |
from swapidseq_inverse_exists[OF n] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
470 |
obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast |
30488 | 471 |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
472 |
from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
473 |
show ?thesis by arith |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
474 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
475 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
476 |
lemma evenperm_unique: assumes p: "swapidseq n p" and n:"even n = b" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
477 |
shows "evenperm p = b" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
478 |
unfolding n[symmetric] evenperm_def |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
479 |
apply (rule swapidseq_even_even[where p = p]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
480 |
apply (rule someI[where x = n]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
481 |
using p by blast+ |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
482 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
483 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
484 |
(* And it has the expected composition properties. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
485 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
486 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
487 |
lemma evenperm_id[simp]: "evenperm id = True" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
488 |
apply (rule evenperm_unique[where n = 0]) by simp_all |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
489 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
490 |
lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
491 |
apply (rule evenperm_unique[where n="if a = b then 0 else 1"]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
492 |
by (simp_all add: swapidseq_swap) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
493 |
|
30488 | 494 |
lemma evenperm_comp: |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
495 |
assumes p: "permutation p" and q:"permutation q" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
496 |
shows "evenperm (p o q) = (evenperm p = evenperm q)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
497 |
proof- |
30488 | 498 |
from p q obtain |
499 |
n m where n: "swapidseq n p" and m: "swapidseq m q" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
500 |
unfolding permutation_def by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
501 |
note nm = swapidseq_comp_add[OF n m] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
502 |
have th: "even (n + m) = (even n \<longleftrightarrow> even m)" by arith |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
503 |
from evenperm_unique[OF n refl] evenperm_unique[OF m refl] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
504 |
evenperm_unique[OF nm th] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
505 |
show ?thesis by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
506 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
507 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
508 |
lemma evenperm_inv: assumes p: "permutation p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
509 |
shows "evenperm (inv p) = evenperm p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
510 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
511 |
from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
512 |
from evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
513 |
show ?thesis . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
514 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
515 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
516 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
517 |
(* A more abstract characterization of permutations. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
518 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
519 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
520 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
521 |
lemma bij_iff: "bij f \<longleftrightarrow> (\<forall>x. \<exists>!y. f y = x)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
522 |
unfolding bij_def inj_on_def surj_def |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
523 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
524 |
apply metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
525 |
apply metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
526 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
527 |
|
30488 | 528 |
lemma permutation_bijective: |
529 |
assumes p: "permutation p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
530 |
shows "bij p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
531 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
532 |
from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast |
30488 | 533 |
from swapidseq_inverse_exists[OF n] obtain q where |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
534 |
q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
535 |
thus ?thesis unfolding bij_iff apply (auto simp add: expand_fun_eq) apply metis done |
30488 | 536 |
qed |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
537 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
538 |
lemma permutation_finite_support: assumes p: "permutation p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
539 |
shows "finite {x. p x \<noteq> x}" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
540 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
541 |
from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
542 |
from n show ?thesis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
543 |
proof(induct n p rule: swapidseq.induct) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
544 |
case id thus ?case by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
545 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
546 |
case (comp_Suc n p a b) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
547 |
let ?S = "insert a (insert b {x. p x \<noteq> x})" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
548 |
from comp_Suc.hyps(2) have fS: "finite ?S" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
549 |
from `a \<noteq> b` have th: "{x. (Fun.swap a b id o p) x \<noteq> x} \<subseteq> ?S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
550 |
by (auto simp add: swap_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
551 |
from finite_subset[OF th fS] show ?case . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
552 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
553 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
554 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
555 |
lemma bij_inv_eq_iff: "bij p ==> x = inv p y \<longleftrightarrow> p x = y" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
556 |
using surj_f_inv_f[of p] inv_f_f[of f] by (auto simp add: bij_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
557 |
|
30488 | 558 |
lemma bij_swap_comp: |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
559 |
assumes bp: "bij p" shows "Fun.swap a b id o p = Fun.swap (inv p a) (inv p b) p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
560 |
using surj_f_inv_f[OF bij_is_surj[OF bp]] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
561 |
by (simp add: expand_fun_eq swap_def bij_inv_eq_iff[OF bp]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
562 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
563 |
lemma bij_swap_ompose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id o p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
564 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
565 |
assume H: "bij p" |
30488 | 566 |
show ?thesis |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
567 |
unfolding bij_swap_comp[OF H] bij_swap_iff |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
568 |
using H . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
569 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
570 |
|
30488 | 571 |
lemma permutation_lemma: |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
572 |
assumes fS: "finite S" and p: "bij p" and pS: "\<forall>x. x\<notin> S \<longrightarrow> p x = x" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
573 |
shows "permutation p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
574 |
using fS p pS |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
575 |
proof(induct S arbitrary: p rule: finite_induct) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
576 |
case (empty p) thus ?case by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
577 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
578 |
case (insert a F p) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
579 |
let ?r = "Fun.swap a (p a) id o p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
580 |
let ?q = "Fun.swap a (p a) id o ?r " |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
581 |
have raa: "?r a = a" by (simp add: swap_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
582 |
from bij_swap_ompose_bij[OF insert(4)] |
30488 | 583 |
have br: "bij ?r" . |
584 |
||
585 |
from insert raa have th: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
586 |
apply (clarsimp simp add: swap_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
587 |
apply (erule_tac x="x" in allE) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
588 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
589 |
unfolding bij_iff apply metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
590 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
591 |
from insert(3)[OF br th] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
592 |
have rp: "permutation ?r" . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
593 |
have "permutation ?q" by (simp add: permutation_compose permutation_swap_id rp) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
594 |
thus ?case by (simp add: o_assoc) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
595 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
596 |
|
30488 | 597 |
lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
598 |
(is "?lhs \<longleftrightarrow> ?b \<and> ?f") |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
599 |
proof |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
600 |
assume p: ?lhs |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
601 |
from p permutation_bijective permutation_finite_support show "?b \<and> ?f" by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
602 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
603 |
assume bf: "?b \<and> ?f" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
604 |
hence bf: "?f" "?b" by blast+ |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
605 |
from permutation_lemma[OF bf] show ?lhs by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
606 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
607 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
608 |
lemma permutation_inverse_works: assumes p: "permutation p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
609 |
shows "inv p o p = id" "p o inv p = id" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
610 |
using permutation_bijective[OF p] surj_iff bij_def inj_iff by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
611 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
612 |
lemma permutation_inverse_compose: |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
613 |
assumes p: "permutation p" and q: "permutation q" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
614 |
shows "inv (p o q) = inv q o inv p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
615 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
616 |
note ps = permutation_inverse_works[OF p] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
617 |
note qs = permutation_inverse_works[OF q] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
618 |
have "p o q o (inv q o inv p) = p o (q o inv q) o inv p" by (simp add: o_assoc) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
619 |
also have "\<dots> = id" by (simp add: ps qs) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
620 |
finally have th0: "p o q o (inv q o inv p) = id" . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
621 |
have "inv q o inv p o (p o q) = inv q o (inv p o p) o q" by (simp add: o_assoc) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
622 |
also have "\<dots> = id" by (simp add: ps qs) |
30488 | 623 |
finally have th1: "inv q o inv p o (p o q) = id" . |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
624 |
from inv_unique_comp[OF th0 th1] show ?thesis . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
625 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
626 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
627 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
628 |
(* Relation to "permutes". *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
629 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
630 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
631 |
lemma permutation_permutes: "permutation p \<longleftrightarrow> (\<exists>S. finite S \<and> p permutes S)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
632 |
unfolding permutation permutes_def bij_iff[symmetric] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
633 |
apply (rule iffI, clarify) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
634 |
apply (rule exI[where x="{x. p x \<noteq> x}"]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
635 |
apply simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
636 |
apply clarsimp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
637 |
apply (rule_tac B="S" in finite_subset) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
638 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
639 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
640 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
641 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
642 |
(* Hence a sort of induction principle composing by swaps. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
643 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
644 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
645 |
lemma permutes_induct: "finite S \<Longrightarrow> P id \<Longrightarrow> (\<And> a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> P p \<Longrightarrow> P p \<Longrightarrow> permutation p ==> P (Fun.swap a b id o p)) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
646 |
==> (\<And>p. p permutes S ==> P p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
647 |
proof(induct S rule: finite_induct) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
648 |
case empty thus ?case by auto |
30488 | 649 |
next |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
650 |
case (insert x F p) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
651 |
let ?r = "Fun.swap x (p x) id o p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
652 |
let ?q = "Fun.swap x (p x) id o ?r" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
653 |
have qp: "?q = p" by (simp add: o_assoc) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
654 |
from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r" by blast |
30488 | 655 |
from permutes_in_image[OF insert.prems(3), of x] |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
656 |
have pxF: "p x \<in> insert x F" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
657 |
have xF: "x \<in> insert x F" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
658 |
have rp: "permutation ?r" |
30488 | 659 |
unfolding permutation_permutes using insert.hyps(1) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
660 |
permutes_insert_lemma[OF insert.prems(3)] by blast |
30488 | 661 |
from insert.prems(2)[OF xF pxF Pr Pr rp] |
662 |
show ?case unfolding qp . |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
663 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
664 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
665 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
666 |
(* Sign of a permutation as a real number. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
667 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
668 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
669 |
definition "sign p = (if evenperm p then (1::int) else -1)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
670 |
|
30488 | 671 |
lemma sign_nz: "sign p \<noteq> 0" by (simp add: sign_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
672 |
lemma sign_id: "sign id = 1" by (simp add: sign_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
673 |
lemma sign_inverse: "permutation p ==> sign (inv p) = sign p" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
674 |
by (simp add: sign_def evenperm_inv) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
675 |
lemma sign_compose: "permutation p \<Longrightarrow> permutation q ==> sign (p o q) = sign(p) * sign(q)" by (simp add: sign_def evenperm_comp) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
676 |
lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else -1)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
677 |
by (simp add: sign_def evenperm_swap) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
678 |
lemma sign_idempotent: "sign p * sign p = 1" by (simp add: sign_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
679 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
680 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
681 |
(* More lemmas about permutations. *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
682 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
683 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
684 |
lemma permutes_natset_le: |
30037 | 685 |
assumes p: "p permutes (S::'a::wellorder set)" and le: "\<forall>i \<in> S. p i <= i" shows "p = id" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
686 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
687 |
{fix n |
30488 | 688 |
have "p n = n" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
689 |
using p le |
30037 | 690 |
proof(induct n arbitrary: S rule: less_induct) |
30488 | 691 |
fix n S assume H: "\<And>m S. \<lbrakk>m < n; p permutes S; \<forall>i\<in>S. p i \<le> i\<rbrakk> \<Longrightarrow> p m = m" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
692 |
"p permutes S" "\<forall>i \<in>S. p i \<le> i" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
693 |
{assume "n \<notin> S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
694 |
with H(2) have "p n = n" unfolding permutes_def by metis} |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
695 |
moreover |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
696 |
{assume ns: "n \<in> S" |
30488 | 697 |
from H(3) ns have "p n < n \<or> p n = n" by auto |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
698 |
moreover{assume h: "p n < n" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
699 |
from H h have "p (p n) = p n" by metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
700 |
with permutes_inj[OF H(2)] have "p n = n" unfolding inj_on_def by blast |
30037 | 701 |
with h have False by simp} |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
702 |
ultimately have "p n = n" by blast } |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
703 |
ultimately show "p n = n" by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
704 |
qed} |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
705 |
thus ?thesis by (auto simp add: expand_fun_eq) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
706 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
707 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
708 |
lemma permutes_natset_ge: |
30037 | 709 |
assumes p: "p permutes (S::'a::wellorder set)" and le: "\<forall>i \<in> S. p i \<ge> i" shows "p = id" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
710 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
711 |
{fix i assume i: "i \<in> S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
712 |
from i permutes_in_image[OF permutes_inv[OF p]] have "inv p i \<in> S" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
713 |
with le have "p (inv p i) \<ge> inv p i" by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
714 |
with permutes_inverses[OF p] have "i \<ge> inv p i" by simp} |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
715 |
then have th: "\<forall>i\<in>S. inv p i \<le> i" by blast |
30488 | 716 |
from permutes_natset_le[OF permutes_inv[OF p] th] |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
717 |
have "inv p = inv id" by simp |
30488 | 718 |
then show ?thesis |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
719 |
apply (subst permutes_inv_inv[OF p, symmetric]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
720 |
apply (rule inv_unique_comp) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
721 |
apply simp_all |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
722 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
723 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
724 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
725 |
lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
726 |
apply (rule set_ext) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
727 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
728 |
using permutes_inv_inv permutes_inv apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
729 |
apply (rule_tac x="inv x" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
730 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
731 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
732 |
|
30488 | 733 |
lemma image_compose_permutations_left: |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
734 |
assumes q: "q permutes S" shows "{q o p | p. p permutes S} = {p . p permutes S}" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
735 |
apply (rule set_ext) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
736 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
737 |
apply (rule permutes_compose) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
738 |
using q apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
739 |
apply (rule_tac x = "inv q o x" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
740 |
by (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
741 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
742 |
lemma image_compose_permutations_right: |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
743 |
assumes q: "q permutes S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
744 |
shows "{p o q | p. p permutes S} = {p . p permutes S}" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
745 |
apply (rule set_ext) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
746 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
747 |
apply (rule permutes_compose) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
748 |
using q apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
749 |
apply (rule_tac x = "x o inv q" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
750 |
by (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o o_assoc[symmetric]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
751 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
752 |
lemma permutes_in_seg: "p permutes {1 ..n} \<Longrightarrow> i \<in> {1..n} ==> 1 <= p i \<and> p i <= n" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
753 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
754 |
apply (simp add: permutes_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
755 |
apply metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
756 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
757 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
758 |
term setsum |
30036 | 759 |
lemma setsum_permutations_inverse: "setsum f {p. p permutes S} = setsum (\<lambda>p. f(inv p)) {p. p permutes S}" (is "?lhs = ?rhs") |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
760 |
proof- |
30036 | 761 |
let ?S = "{p . p permutes S}" |
30488 | 762 |
have th0: "inj_on inv ?S" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
763 |
proof(auto simp add: inj_on_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
764 |
fix q r |
30036 | 765 |
assume q: "q permutes S" and r: "r permutes S" and qr: "inv q = inv r" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
766 |
hence "inv (inv q) = inv (inv r)" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
767 |
with permutes_inv_inv[OF q] permutes_inv_inv[OF r] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
768 |
show "q = r" by metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
769 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
770 |
have th1: "inv ` ?S = ?S" using image_inverse_permutations by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
771 |
have th2: "?rhs = setsum (f o inv) ?S" by (simp add: o_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
772 |
from setsum_reindex[OF th0, of f] show ?thesis unfolding th1 th2 . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
773 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
774 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
775 |
lemma setum_permutations_compose_left: |
30036 | 776 |
assumes q: "q permutes S" |
777 |
shows "setsum f {p. p permutes S} = |
|
778 |
setsum (\<lambda>p. f(q o p)) {p. p permutes S}" (is "?lhs = ?rhs") |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
779 |
proof- |
30036 | 780 |
let ?S = "{p. p permutes S}" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
781 |
have th0: "?rhs = setsum (f o (op o q)) ?S" by (simp add: o_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
782 |
have th1: "inj_on (op o q) ?S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
783 |
apply (auto simp add: inj_on_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
784 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
785 |
fix p r |
30036 | 786 |
assume "p permutes S" and r:"r permutes S" and rp: "q \<circ> p = q \<circ> r" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
787 |
hence "inv q o q o p = inv q o q o r" by (simp add: o_assoc[symmetric]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
788 |
with permutes_inj[OF q, unfolded inj_iff] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
789 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
790 |
show "p = r" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
791 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
792 |
have th3: "(op o q) ` ?S = ?S" using image_compose_permutations_left[OF q] by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
793 |
from setsum_reindex[OF th1, of f] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
794 |
show ?thesis unfolding th0 th1 th3 . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
795 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
796 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
797 |
lemma sum_permutations_compose_right: |
30036 | 798 |
assumes q: "q permutes S" |
799 |
shows "setsum f {p. p permutes S} = |
|
800 |
setsum (\<lambda>p. f(p o q)) {p. p permutes S}" (is "?lhs = ?rhs") |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
801 |
proof- |
30036 | 802 |
let ?S = "{p. p permutes S}" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
803 |
have th0: "?rhs = setsum (f o (\<lambda>p. p o q)) ?S" by (simp add: o_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
804 |
have th1: "inj_on (\<lambda>p. p o q) ?S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
805 |
apply (auto simp add: inj_on_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
806 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
807 |
fix p r |
30036 | 808 |
assume "p permutes S" and r:"r permutes S" and rp: "p o q = r o q" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
809 |
hence "p o (q o inv q) = r o (q o inv q)" by (simp add: o_assoc) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
810 |
with permutes_surj[OF q, unfolded surj_iff] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
811 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
812 |
show "p = r" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
813 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
814 |
have th3: "(\<lambda>p. p o q) ` ?S = ?S" using image_compose_permutations_right[OF q] by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
815 |
from setsum_reindex[OF th1, of f] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
816 |
show ?thesis unfolding th0 th1 th3 . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
817 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
818 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
819 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
820 |
(* Sum over a set of permutations (could generalize to iteration). *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
821 |
(* ------------------------------------------------------------------------- *) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
822 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
823 |
lemma setsum_over_permutations_insert: |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
824 |
assumes fS: "finite S" and aS: "a \<notin> S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
825 |
shows "setsum f {p. p permutes (insert a S)} = setsum (\<lambda>b. setsum (\<lambda>q. f (Fun.swap a b id o q)) {p. p permutes S}) (insert a S)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
826 |
proof- |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
827 |
have th0: "\<And>f a b. (\<lambda>(b,p). f (Fun.swap a b id o p)) = f o (\<lambda>(b,p). Fun.swap a b id o p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
828 |
by (simp add: expand_fun_eq) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
829 |
have th1: "\<And>P Q. P \<times> Q = {(a,b). a \<in> P \<and> b \<in> Q}" by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
830 |
have th2: "\<And>P Q. P \<Longrightarrow> (P \<Longrightarrow> Q) \<Longrightarrow> P \<and> Q" by blast |
30488 | 831 |
show ?thesis |
832 |
unfolding permutes_insert |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
833 |
unfolding setsum_cartesian_product |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
834 |
unfolding th1[symmetric] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
835 |
unfolding th0 |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
836 |
proof(rule setsum_reindex) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
837 |
let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
838 |
let ?P = "{p. p permutes S}" |
30488 | 839 |
{fix b c p q assume b: "b \<in> insert a S" and c: "c \<in> insert a S" |
840 |
and p: "p permutes S" and q: "q permutes S" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
841 |
and eq: "Fun.swap a b id o p = Fun.swap a c id o q" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
842 |
from p q aS have pa: "p a = a" and qa: "q a = a" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
843 |
unfolding permutes_def by metis+ |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
844 |
from eq have "(Fun.swap a b id o p) a = (Fun.swap a c id o q) a" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
845 |
hence bc: "b = c" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
846 |
apply (simp add: permutes_def pa qa o_def fun_upd_def swap_def id_def cong del: if_weak_cong) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
847 |
apply (cases "a = b", auto) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
848 |
by (cases "b = c", auto) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
849 |
from eq[unfolded bc] have "(\<lambda>p. Fun.swap a c id o p) (Fun.swap a c id o p) = (\<lambda>p. Fun.swap a c id o p) (Fun.swap a c id o q)" by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
850 |
hence "p = q" unfolding o_assoc swap_id_idempotent |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
851 |
by (simp add: o_def) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
852 |
with bc have "b = c \<and> p = q" by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
853 |
} |
30488 | 854 |
|
855 |
then show "inj_on ?f (insert a S \<times> ?P)" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
856 |
unfolding inj_on_def |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
857 |
apply clarify by metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
858 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
859 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
860 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
861 |
end |