Theory "sum_num"

Parents rich_list

Signature

Constant	Type
GSUM	:num # num -> (num -> num) -> num
GSUM_tupled	:(num # num) # (num -> num) -> num
SUM	:num -> (num -> num) -> num

Definitions

GSUM_tupled_primitive_def

|- GSUM_tupled =
   WFREC (@R. WF R ∧ ∀f m n. R ((n,m),f) ((n,SUC m),f))
     (λGSUM_tupled a.
        case a of
          ((n,0),f) => I 0
        | ((n,SUC m),f) => I (GSUM_tupled ((n,m),f) + f (n + m)))

GSUM_curried_def

|- ∀x x1. GSUM x x1 = GSUM_tupled (x,x1)

SUM_def

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

Theorems

GSUM_ind

|- ∀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

GSUM_def

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

GSUM_def_compute

|- (∀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))

GSUM_1

|- ∀m f. GSUM (m,1) f = f m

GSUM_ADD

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

GSUM_ZERO

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

GSUM_MONO

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

GSUM_LESS

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

GSUM_EQUAL

|- ∀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)

GSUM_FUN_EQUAL

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

SUM_def_compute

|- (∀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))

SUM

|- ∀m f. SUM m f = GSUM (0,m) f

SUM_1

|- ∀f. SUM 1 f = f 0

SUM_MONO

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

SUM_LESS

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

SUM_EQUAL

|- ∀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)

SUM_FUN_EQUAL

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

SUM_ZERO

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

SUM_FOLDL

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