The Coq Proof Assistant

# Library Coq.Reals.Binomial

Require Import Rbase.
Require Import Rfunctions.
Require Import PartSum.
Local Open Scope R_scope.
Definition C (n p:nat) : R :=
INR (fact n) / (INR (fact p) * INR (fact (n - p))).
Lemma pascal_step1 : forall n i:nat, (i <= n)%nat -> C n i = C n (n - i).
Lemma pascal_step2 :
forall n i:nat,
(i <= n)%nat -> C (S n) i = INR (S n) / INR (S n - i) * C n i.
Lemma pascal_step3 :
forall n i:nat, (i < n)%nat -> C n (S i) = INR (n - i) / INR (S i) * C n i.
Lemma pascal :
forall n i:nat, (i < n)%nat -> C n i + C n (S i) = C (S n) (S i).
Lemma binomial :
forall (x y:R) (n:nat),
(x + y) ^ n = sum_f_R0 (fun i:nat => C n i * x ^ i * y ^ (n - i)) n.