Theory sum_num

⊢ (∀f. SUM 0 f = 0) ∧ ∀m f. SUM (SUC m) f = SUM m f + f m

⊢ ∀m f. GSUM (m,1) f = f m

⊢ ∀p m n f. GSUM (p,m + n) f = GSUM (p,m) f + GSUM (p + m,n) f

⊢ ∀p m n f.
    GSUM (p,m) f = GSUM (p,n) f ⇔
    m ≤ n ∧ (∀q. p + m ≤ q ∧ q < p + n ⇒ f q = 0) ∨
    n < m ∧ ∀q. p + n ≤ q ∧ q < p + m ⇒ f q = 0

⊢ ∀p n f g.
    (∀x. p ≤ x ∧ x < p + n ⇒ f x = g x) ⇒ GSUM (p,n) f = GSUM (p,n) g

⊢ ∀p m n f.
    (∃q. m + p ≤ q ∧ q < n + p ∧ f q ≠ 0) ⇔ GSUM (p,m) f < GSUM (p,n) f

⊢ ∀p m n f. m ≤ n ∧ f (p + n) ≠ 0 ⇒ GSUM (p,m) f < GSUM (p,SUC n) f

⊢ ∀p n f. (∀m. p ≤ m ∧ m < p + n ⇒ f m = 0) ⇔ GSUM (p,n) f = 0

⊢ (∀n f. GSUM (n,0) f = 0) ∧
  (∀n m f.
     GSUM (n,NUMERAL (BIT1 m)) f =
     GSUM (n,NUMERAL (BIT1 m) − 1) f + f (n + (NUMERAL (BIT1 m) − 1))) ∧
  ∀n m f.
    GSUM (n,NUMERAL (BIT2 m)) f =
    GSUM (n,NUMERAL (BIT1 m)) f + f (n + NUMERAL (BIT1 m))

⊢ (∀n f. GSUM (n,0) f = 0) ∧
  ∀n m f. GSUM (n,SUC m) f = GSUM (n,m) f + f (n + m)

⊢ ∀P. (∀n f. P (n,0) f) ∧ (∀n m f. P (n,m) f ⇒ P (n,SUC m) f) ⇒
      ∀v v1 v2. P (v,v1) v2

⊢ ∀m f. SUM m f = GSUM (0,m) f

⊢ ∀f. SUM 1 f = f 0

⊢ ∀m n f.
    SUM m f = SUM n f ⇔
    m ≤ n ∧ (∀q. m ≤ q ∧ q < n ⇒ f q = 0) ∨
    n < m ∧ ∀q. n ≤ q ∧ q < m ⇒ f q = 0

⊢ ∀n f. SUM n f = FOLDL (λx n. f n + x) 0 (COUNT_LIST n)

⊢ ∀f g. (∀x. x < n ⇒ f x = g x) ⇒ SUM n f = SUM n g

⊢ ∀m n f. (∃q. m ≤ q ∧ q < n ∧ f q ≠ 0) ⇔ SUM m f < SUM n f

⊢ ∀m n f. m ≤ n ∧ f n ≠ 0 ⇒ SUM m f < SUM (SUC n) f

⊢ ∀n f. (∀m. m < n ⇒ f m = 0) ⇔ SUM n f = 0

⊢ (∀f. SUM 0 f = 0) ∧
  (∀m f.
     SUM (NUMERAL (BIT1 m)) f =
     SUM (NUMERAL (BIT1 m) − 1) f + f (NUMERAL (BIT1 m) − 1)) ∧
  ∀m f.
    SUM (NUMERAL (BIT2 m)) f =
    SUM (NUMERAL (BIT1 m)) f + f (NUMERAL (BIT1 m))