Structure sum_numTheory

signature sum_numTheory =
sig
  type thm = Thm.thm

  (*  Definitions  *)
    val GSUM_curried_def : thm
    val GSUM_tupled_primitive_def : thm
    val SUM_def : thm

  (*  Theorems  *)
    val GSUM_1 : thm
    val GSUM_ADD : thm
    val GSUM_EQUAL : thm
    val GSUM_FUN_EQUAL : thm
    val GSUM_LESS : thm
    val GSUM_MONO : thm
    val GSUM_ZERO : thm
    val GSUM_def : thm
    val GSUM_def_compute : thm
    val GSUM_ind : thm
    val SUM : thm
    val SUM_1 : thm
    val SUM_EQUAL : thm
    val SUM_FOLDL : thm
    val SUM_FUN_EQUAL : thm
    val SUM_LESS : thm
    val SUM_MONO : thm
    val SUM_ZERO : thm
    val SUM_def_compute : thm

  val sum_num_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [rich_list] Parent theory of "sum_num"

   [GSUM_curried_def]  Definition

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

   [GSUM_tupled_primitive_def]  Definition

      |- 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)))

   [SUM_def]  Definition

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

   [GSUM_1]  Theorem

      |- ∀m f. GSUM (m,1) f = f m

   [GSUM_ADD]  Theorem

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

   [GSUM_EQUAL]  Theorem

      |- ∀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]  Theorem

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

   [GSUM_LESS]  Theorem

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

   [GSUM_MONO]  Theorem

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

   [GSUM_ZERO]  Theorem

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

   [GSUM_def]  Theorem

      |- (∀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]  Theorem

      |- (∀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_ind]  Theorem

      |- ∀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

   [SUM]  Theorem

      |- ∀m f. SUM m f = GSUM (0,m) f

   [SUM_1]  Theorem

      |- ∀f. SUM 1 f = f 0

   [SUM_EQUAL]  Theorem

      |- ∀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_FOLDL]  Theorem

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

   [SUM_FUN_EQUAL]  Theorem

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

   [SUM_LESS]  Theorem

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

   [SUM_MONO]  Theorem

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

   [SUM_ZERO]  Theorem

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

   [SUM_def_compute]  Theorem

      |- (∀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))


*)
end

HOL 4, Kananaskis-10