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module SumSingle
use int.Int
use real.RealInfix
use real.Abs
use real.Sum
use ufloat.USingle
use ufloat.USingleLemmas
use ref.Ref
use real.FromInt
function abs_real_fun (f:int -> usingle) : int -> real =
fun i -> abs (to_real (f i))
let rec lemma sum_of_fun_le_sum_of_abs_fun (f : int -> usingle) (i:int)
requires { 0 <= i }
variant { i }
ensures { abs (sum (real_fun f) 0 i) <=. sum (abs_real_fun f) 0 i }
= if i > 1 then sum_of_fun_le_sum_of_abs_fun f (i - 1)
let ghost usum (f : int -> usingle) (n:int) : usingle
requires { 0 <= n }
ensures {
abs (to_real result -. sum (real_fun f) 0 n) <=. sum (abs_real_fun f) 0 n *. (eps *. from_int n)
}
=
let s = ref u0 in
for i = 0 to n - 1 do
(* We use the propagation lemma on IEEE addition to prove the invariant *)
invariant {
abs (to_real !s -. sum (real_fun f) 0 i) <=. sum (abs_real_fun f) 0 i *. (eps *. from_int i)
}
s := !s ++. f i;
done;
!s
(* We use the propagation lemma on IEEE addition to prove the postcondition *)
let rec usum_rec (f : int -> usingle) (n:int) : usingle
requires { 0 <= n }
variant { n }
ensures {
abs (to_real result -. sum (real_fun f) 0 n) <=. sum (abs_real_fun f) 0 n *. (eps *. from_int n)
}
= if n = 0 then uzero else usum_rec f (n-1) ++. f (n-1)
function exact_f : int -> real
function f' : int -> real
axiom f'_def : f' = fun i -> abs (exact_f i)
constant f_rel_err:real
constant f_cst_err:real
(* We use the propagation on sum to prove the postcondition *)
let function example1 (f : int -> usingle) (n : int) : usingle
requires { 0 <= n }
requires { forall i. abs (to_real (f i) -. exact_f i) <=. f' i *. f_rel_err }
ensures {
abs (to_real result -. sum exact_f 0 n)
<=. sum f' 0 n *. (f_rel_err +. (eps *. from_int n) *. (1. +. f_rel_err))
}
= usum_rec f n
end
module SumDouble
use int.Int
use real.RealInfix
use real.Abs
use real.Sum
use ufloat.UDouble
use ufloat.UDoubleLemmas
use ref.Ref
use real.FromInt
function abs_real_fun (f:int -> udouble) : int -> real =
fun i -> abs (to_real (f i))
let rec lemma sum_of_fun_le_sum_of_abs_fun (f : int -> udouble) (i:int)
requires { 0 <= i }
variant { i }
ensures { abs (sum (real_fun f) 0 i) <=. sum (abs_real_fun f) 0 i }
= if i > 1 then sum_of_fun_le_sum_of_abs_fun f (i - 1)
let ghost usum (f : int -> udouble) (n:int) : udouble
requires { 0 <= n }
ensures {
abs (to_real result -. sum (real_fun f) 0 n) <=. sum (abs_real_fun f) 0 n *. (eps *. from_int n)
}
=
let s = ref u0 in
for i = 0 to n - 1 do
(* We use the propagation lemma on IEEE addition to prove the invariant *)
invariant {
abs (to_real !s -. sum (real_fun f) 0 i) <=. sum (abs_real_fun f) 0 i *. (eps *. from_int i)
}
s := !s ++. f i;
done;
!s
(* We use the propagation lemma on IEEE addition to prove the postcondition *)
let rec usum_rec (f : int -> udouble) (n:int) : udouble
requires { 0 <= n }
variant { n }
ensures {
abs (to_real result -. sum (real_fun f) 0 n) <=. sum (abs_real_fun f) 0 n *. (eps *. from_int n)
}
= if n = 0 then uzero else usum_rec f (n-1) ++. f (n-1)
function exact_f : int -> real
function f' : int -> real
axiom f'_def : f' = fun i -> abs (exact_f i)
constant f_rel_err:real
constant f_cst_err:real
(* We use the propagation on sum to prove the postcondition *)
let function example1 (f : int -> udouble) (n : int) : udouble
requires { 0 <= n }
requires { forall i. abs (to_real (f i) -. exact_f i) <=. f' i *. f_rel_err }
ensures {
abs (to_real result -. sum exact_f 0 n)
<=. sum f' 0 n *. (f_rel_err +. (eps *. from_int n) *. (1. +. f_rel_err))
}
= usum_rec f n
end
|
