(* Title: HOL/NatArith.thy
ID: $Id$
Author: Tobias Nipkow and Markus Wenzel
License: GPL (GNU GENERAL PUBLIC LICENSE)
*)
header {* More arithmetic on natural numbers *}
theory NatArith = Nat
files "arith_data.ML":
setup arith_setup
lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m <= n)"
by (simp add: less_eq reflcl_trancl [symmetric]
del: reflcl_trancl, arith)
lemma nat_diff_split:
"P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
-- {* elimination of @{text -} on @{text nat} *}
by (cases "a<b" rule: case_split)
(auto simp add: diff_is_0_eq [THEN iffD2])
lemma nat_diff_split_asm:
"P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
-- {* elimination of @{text -} on @{text nat} in assumptions *}
by (simp split: nat_diff_split)
ML {*
val nat_diff_split = thm "nat_diff_split";
val nat_diff_split_asm = thm "nat_diff_split_asm";
*}
(* Careful: arith_tac produces counter examples!
fun add_arith cs = cs addafter ("arith_tac", arith_tac);
TODO: use arith_tac for force_tac in Provers/clasimp.ML *)
lemmas [arith_split] = nat_diff_split split_min split_max
subsection {* Generic summation indexed over natural numbers *}
consts
Summation :: "(nat => 'a::{zero, plus}) => nat => 'a"
primrec
"Summation f 0 = 0"
"Summation f (Suc n) = Summation f n + f n"
syntax
"_Summation" :: "idt => nat => 'a => nat" ("\<Sum>_<_. _" [0, 51, 10] 10)
translations
"\<Sum>i < n. b" == "Summation (\<lambda>i. b) n"
theorem Summation_step:
"0 < n ==> (\<Sum>i < n. f i) = (\<Sum>i < n - 1. f i) + f (n - 1)"
by (induct n) simp_all
end