(* Title: HOL/SPARK/Examples/RIPEMD-160/RMD_Lemmas.thy
Author: Fabian Immler, TU Muenchen
Verification of the RIPEMD-160 hash function
*)
theory RMD_Lemmas
imports Main
begin
definition "fun_of_list i xs g j =
(if j < i \<or> i + int (length xs) \<le> j then g j else xs ! nat (j - i))"
lemma fun_of_list_Nil [simp]: "fun_of_list i [] g = g"
by (auto simp add: fun_eq_iff fun_of_list_def)
lemma fun_of_list_Cons [simp]:
"fun_of_list i (x # xs) g = fun_of_list (i + 1) xs (g(i:=x))"
by (auto simp add: fun_eq_iff fun_of_list_def nth_Cons'
nat_diff_distrib [of "int (Suc 0)", simplified, symmetric]
diff_diff_eq)
lemma nth_fun_of_list_eq:
"0 \<le> i \<Longrightarrow> i < int (length xs) \<Longrightarrow> xs ! nat i = fun_of_list 0 xs g i"
by (simp add: fun_of_list_def)
end