(* Title: HOL/Relation_Power.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1996 TU Muenchen
*)
header{*Powers of Relations and Functions*}
theory Relation_Power
imports Nat
begin
instance
set :: (type) power ..
--{* only type @{typ "('a * 'a) set"} should be in class @{text power}!*}
(*R^n = R O ... O R, the n-fold composition of R*)
primrec (relpow)
"R^0 = Id"
"R^(Suc n) = R O (R^n)"
instance
fun :: (type, type) power ..
--{* only type @{typ "'a => 'a"} should be in class @{text power}!*}
(*f^n = f o ... o f, the n-fold composition of f*)
primrec (funpow)
"f^0 = id"
"f^(Suc n) = f o (f^n)"
text{*WARNING: due to the limits of Isabelle's type classes, exponentiation on
functions and relations has too general a domain, namely @{typ "('a * 'b)set"}
and @{typ "'a => 'b"}. Explicit type constraints may therefore be necessary.
For example, @{term "range(f^n) = A"} and @{term "Range(R^n) = B"} need
constraints.*}
lemma funpow_add: "f ^ (m+n) = f^m o f^n"
by (induct m) simp_all
lemma funpow_swap1: "f((f^n) x) = (f^n)(f x)"
proof -
have "f((f^n) x) = (f^(n+1)) x" by simp
also have "\<dots> = (f^n o f^1) x" by (simp only: funpow_add)
also have "\<dots> = (f^n)(f x)" by simp
finally show ?thesis .
qed
lemma rel_pow_1 [simp]:
fixes R :: "('a*'a)set"
shows "R^1 = R"
by simp
lemma rel_pow_0_I: "(x,x) : R^0"
by simp
lemma rel_pow_Suc_I: "[| (x,y) : R^n; (y,z):R |] ==> (x,z):R^(Suc n)"
by auto
lemma rel_pow_Suc_I2:
"(x, y) : R \<Longrightarrow> (y, z) : R^n \<Longrightarrow> (x,z) : R^(Suc n)"
apply (induct n fixing: z)
apply simp
apply fastsimp
done
lemma rel_pow_0_E: "[| (x,y) : R^0; x=y ==> P |] ==> P"
by simp
lemma rel_pow_Suc_E:
"[| (x,z) : R^(Suc n); !!y. [| (x,y) : R^n; (y,z) : R |] ==> P |] ==> P"
by auto
lemma rel_pow_E:
"[| (x,z) : R^n; [| n=0; x = z |] ==> P;
!!y m. [| n = Suc m; (x,y) : R^m; (y,z) : R |] ==> P
|] ==> P"
by (cases n) auto
lemma rel_pow_Suc_D2:
"(x, z) : R^(Suc n) \<Longrightarrow> (\<exists>y. (x,y) : R & (y,z) : R^n)"
apply (induct n fixing: x z)
apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
done
lemma rel_pow_Suc_D2':
"\<forall>x y z. (x,y) : R^n & (y,z) : R --> (\<exists>w. (x,w) : R & (w,z) : R^n)"
by (induct n) (simp_all, blast)
lemma rel_pow_E2:
"[| (x,z) : R^n; [| n=0; x = z |] ==> P;
!!y m. [| n = Suc m; (x,y) : R; (y,z) : R^m |] ==> P
|] ==> P"
apply (case_tac n, simp)
apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)
done
lemma rtrancl_imp_UN_rel_pow: "!!p. p:R^* ==> p : (UN n. R^n)"
apply (simp only: split_tupled_all)
apply (erule rtrancl_induct)
apply (blast intro: rel_pow_0_I rel_pow_Suc_I)+
done
lemma rel_pow_imp_rtrancl: "!!p. p:R^n ==> p:R^*"
apply (simp only: split_tupled_all)
apply (induct n)
apply (blast intro: rtrancl_refl elim: rel_pow_0_E)
apply (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
done
lemma rtrancl_is_UN_rel_pow: "R^* = (UN n. R^n)"
by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
lemma single_valued_rel_pow:
"!!r::('a * 'a)set. single_valued r ==> single_valued (r^n)"
apply (rule single_valuedI)
apply (induct n)
apply simp
apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
done
ML
{*
val funpow_add = thm "funpow_add";
val rel_pow_1 = thm "rel_pow_1";
val rel_pow_0_I = thm "rel_pow_0_I";
val rel_pow_Suc_I = thm "rel_pow_Suc_I";
val rel_pow_Suc_I2 = thm "rel_pow_Suc_I2";
val rel_pow_0_E = thm "rel_pow_0_E";
val rel_pow_Suc_E = thm "rel_pow_Suc_E";
val rel_pow_E = thm "rel_pow_E";
val rel_pow_Suc_D2 = thm "rel_pow_Suc_D2";
val rel_pow_Suc_D2 = thm "rel_pow_Suc_D2";
val rel_pow_E2 = thm "rel_pow_E2";
val rtrancl_imp_UN_rel_pow = thm "rtrancl_imp_UN_rel_pow";
val rel_pow_imp_rtrancl = thm "rel_pow_imp_rtrancl";
val rtrancl_is_UN_rel_pow = thm "rtrancl_is_UN_rel_pow";
val single_valued_rel_pow = thm "single_valued_rel_pow";
*}
end