Theory "numRing"

Signature

Definitions

num_interp_sp_def

|- num_interp_sp = interp_sp (semi_ring 0 1 $+ $* )

num_spolynom_simplify_def

|- num_spolynom_simplify = spolynom_simplify (semi_ring 0 1 $+ $* )

num_spolynom_normalize_def

|- num_spolynom_normalize = spolynom_normalize (semi_ring 0 1 $+ $* )

num_interp_cs_def

|- num_interp_cs = interp_cs (semi_ring 0 1 $+ $* )

num_ics_aux_def

|- num_ics_aux = ics_aux (semi_ring 0 1 $+ $* )

num_interp_m_def

|- num_interp_m = interp_m (semi_ring 0 1 $+ $* )

num_interp_vl_def

|- num_interp_vl = interp_vl (semi_ring 0 1 $+ $* )

num_ivl_aux_def

|- num_ivl_aux = ivl_aux (semi_ring 0 1 $+ $* )

num_canonical_sum_simplify_def

|- num_canonical_sum_simplify = canonical_sum_simplify (semi_ring 0 1 $+ $* )

num_canonical_sum_prod_def

|- num_canonical_sum_prod = canonical_sum_prod (semi_ring 0 1 $+ $* )

num_canonical_sum_scalar3_def

|- num_canonical_sum_scalar3 = canonical_sum_scalar3 (semi_ring 0 1 $+ $* )

num_canonical_sum_scalar2_def

|- num_canonical_sum_scalar2 = canonical_sum_scalar2 (semi_ring 0 1 $+ $* )

num_canonical_sum_scalar_def

|- num_canonical_sum_scalar = canonical_sum_scalar (semi_ring 0 1 $+ $* )

num_varlist_insert_def

|- num_varlist_insert = varlist_insert (semi_ring 0 1 $+ $* )

num_monom_insert_def

|- num_monom_insert = monom_insert (semi_ring 0 1 $+ $* )

num_canonical_sum_merge_def

|- num_canonical_sum_merge = canonical_sum_merge (semi_ring 0 1 $+ $* )

Theorems

num_semi_ring

|- is_semi_ring (semi_ring 0 1 $+ $* )

num_ring_thms

|- is_semi_ring (semi_ring 0 1 $+ $* ) ∧
   (∀vm p. num_interp_sp vm p = num_interp_cs vm (num_spolynom_simplify p)) ∧
   (((∀vm c. num_interp_sp vm (SPconst c) = c) ∧
     (∀vm i. num_interp_sp vm (SPvar i) = varmap_find i vm) ∧
     (∀vm p1 p2.
        num_interp_sp vm (SPplus p1 p2) =
        num_interp_sp vm p1 + num_interp_sp vm p2) ∧
     ∀vm p1 p2.
       num_interp_sp vm (SPmult p1 p2) =
       num_interp_sp vm p1 * num_interp_sp vm p2) ∧
    (∀x v2 v1. varmap_find End_idx (Node_vm x v1 v2) = x) ∧
    (∀x v2 v1 i1.
       varmap_find (Right_idx i1) (Node_vm x v1 v2) = varmap_find i1 v2) ∧
    (∀x v2 v1 i1.
       varmap_find (Left_idx i1) (Node_vm x v1 v2) = varmap_find i1 v1) ∧
    ∀i. varmap_find i Empty_vm = @x. T) ∧
   ((∀t2 t1 l2 l1 c2 c1.
       num_canonical_sum_merge (Cons_monom c1 l1 t1) (Cons_monom c2 l2 t2) =
       compare (list_compare index_compare l1 l2)
         (Cons_monom c1 l1 (num_canonical_sum_merge t1 (Cons_monom c2 l2 t2)))
         (Cons_monom (c1 + c2) l1 (num_canonical_sum_merge t1 t2))
         (Cons_monom c2 l2
            (num_canonical_sum_merge (Cons_monom c1 l1 t1) t2))) ∧
    (∀t2 t1 l2 l1 c1.
       num_canonical_sum_merge (Cons_monom c1 l1 t1) (Cons_varlist l2 t2) =
       compare (list_compare index_compare l1 l2)
         (Cons_monom c1 l1 (num_canonical_sum_merge t1 (Cons_varlist l2 t2)))
         (Cons_monom (c1 + 1) l1 (num_canonical_sum_merge t1 t2))
         (Cons_varlist l2
            (num_canonical_sum_merge (Cons_monom c1 l1 t1) t2))) ∧
    (∀t2 t1 l2 l1 c2.
       num_canonical_sum_merge (Cons_varlist l1 t1) (Cons_monom c2 l2 t2) =
       compare (list_compare index_compare l1 l2)
         (Cons_varlist l1 (num_canonical_sum_merge t1 (Cons_monom c2 l2 t2)))
         (Cons_monom (1 + c2) l1 (num_canonical_sum_merge t1 t2))
         (Cons_monom c2 l2
            (num_canonical_sum_merge (Cons_varlist l1 t1) t2))) ∧
    (∀t2 t1 l2 l1.
       num_canonical_sum_merge (Cons_varlist l1 t1) (Cons_varlist l2 t2) =
       compare (list_compare index_compare l1 l2)
         (Cons_varlist l1 (num_canonical_sum_merge t1 (Cons_varlist l2 t2)))
         (Cons_monom (1 + 1) l1 (num_canonical_sum_merge t1 t2))
         (Cons_varlist l2
            (num_canonical_sum_merge (Cons_varlist l1 t1) t2))) ∧
    (∀s1. num_canonical_sum_merge s1 Nil_monom = s1) ∧
    (∀v6 v5 v4.
       num_canonical_sum_merge Nil_monom (Cons_monom v4 v5 v6) =
       Cons_monom v4 v5 v6) ∧
    ∀v8 v7.
      num_canonical_sum_merge Nil_monom (Cons_varlist v7 v8) =
      Cons_varlist v7 v8) ∧
   ((∀t2 l2 l1 c2 c1.
       num_monom_insert c1 l1 (Cons_monom c2 l2 t2) =
       compare (list_compare index_compare l1 l2)
         (Cons_monom c1 l1 (Cons_monom c2 l2 t2)) (Cons_monom (c1 + c2) l1 t2)
         (Cons_monom c2 l2 (num_monom_insert c1 l1 t2))) ∧
    (∀t2 l2 l1 c1.
       num_monom_insert c1 l1 (Cons_varlist l2 t2) =
       compare (list_compare index_compare l1 l2)
         (Cons_monom c1 l1 (Cons_varlist l2 t2)) (Cons_monom (c1 + 1) l1 t2)
         (Cons_varlist l2 (num_monom_insert c1 l1 t2))) ∧
    ∀l1 c1. num_monom_insert c1 l1 Nil_monom = Cons_monom c1 l1 Nil_monom) ∧
   ((∀t2 l2 l1 c2.
       num_varlist_insert l1 (Cons_monom c2 l2 t2) =
       compare (list_compare index_compare l1 l2)
         (Cons_varlist l1 (Cons_monom c2 l2 t2)) (Cons_monom (1 + c2) l1 t2)
         (Cons_monom c2 l2 (num_varlist_insert l1 t2))) ∧
    (∀t2 l2 l1.
       num_varlist_insert l1 (Cons_varlist l2 t2) =
       compare (list_compare index_compare l1 l2)
         (Cons_varlist l1 (Cons_varlist l2 t2)) (Cons_monom (1 + 1) l1 t2)
         (Cons_varlist l2 (num_varlist_insert l1 t2))) ∧
    ∀l1. num_varlist_insert l1 Nil_monom = Cons_varlist l1 Nil_monom) ∧
   ((∀c0 c l t.
       num_canonical_sum_scalar c0 (Cons_monom c l t) =
       Cons_monom (c0 * c) l (num_canonical_sum_scalar c0 t)) ∧
    (∀c0 l t.
       num_canonical_sum_scalar c0 (Cons_varlist l t) =
       Cons_monom c0 l (num_canonical_sum_scalar c0 t)) ∧
    ∀c0. num_canonical_sum_scalar c0 Nil_monom = Nil_monom) ∧
   ((∀l0 c l t.
       num_canonical_sum_scalar2 l0 (Cons_monom c l t) =
       num_monom_insert c (list_merge index_lt l0 l)
         (num_canonical_sum_scalar2 l0 t)) ∧
    (∀l0 l t.
       num_canonical_sum_scalar2 l0 (Cons_varlist l t) =
       num_varlist_insert (list_merge index_lt l0 l)
         (num_canonical_sum_scalar2 l0 t)) ∧
    ∀l0. num_canonical_sum_scalar2 l0 Nil_monom = Nil_monom) ∧
   ((∀c0 l0 c l t.
       num_canonical_sum_scalar3 c0 l0 (Cons_monom c l t) =
       num_monom_insert (c0 * c) (list_merge index_lt l0 l)
         (num_canonical_sum_scalar3 c0 l0 t)) ∧
    (∀c0 l0 l t.
       num_canonical_sum_scalar3 c0 l0 (Cons_varlist l t) =
       num_monom_insert c0 (list_merge index_lt l0 l)
         (num_canonical_sum_scalar3 c0 l0 t)) ∧
    ∀c0 l0. num_canonical_sum_scalar3 c0 l0 Nil_monom = Nil_monom) ∧
   ((∀c1 l1 t1 s2.
       num_canonical_sum_prod (Cons_monom c1 l1 t1) s2 =
       num_canonical_sum_merge (num_canonical_sum_scalar3 c1 l1 s2)
         (num_canonical_sum_prod t1 s2)) ∧
    (∀l1 t1 s2.
       num_canonical_sum_prod (Cons_varlist l1 t1) s2 =
       num_canonical_sum_merge (num_canonical_sum_scalar2 l1 s2)
         (num_canonical_sum_prod t1 s2)) ∧
    ∀s2. num_canonical_sum_prod Nil_monom s2 = Nil_monom) ∧
   ((∀c l t.
       num_canonical_sum_simplify (Cons_monom c l t) =
       if c = 0 then num_canonical_sum_simplify t
       else if c = 1 then Cons_varlist l (num_canonical_sum_simplify t)
       else Cons_monom c l (num_canonical_sum_simplify t)) ∧
    (∀l t.
       num_canonical_sum_simplify (Cons_varlist l t) =
       Cons_varlist l (num_canonical_sum_simplify t)) ∧
    (num_canonical_sum_simplify Nil_monom = Nil_monom)) ∧
   ((∀vm x. num_ivl_aux vm x [] = varmap_find x vm) ∧
    ∀vm x x' t'.
      num_ivl_aux vm x (x'::t') = varmap_find x vm * num_ivl_aux vm x' t') ∧
   ((∀vm. num_interp_vl vm [] = 1) ∧
    ∀vm x t. num_interp_vl vm (x::t) = num_ivl_aux vm x t) ∧
   ((∀vm c. num_interp_m vm c [] = c) ∧
    ∀vm c x t. num_interp_m vm c (x::t) = c * num_ivl_aux vm x t) ∧
   ((∀vm a. num_ics_aux vm a Nil_monom = a) ∧
    (∀vm a l t.
       num_ics_aux vm a (Cons_varlist l t) =
       a + num_ics_aux vm (num_interp_vl vm l) t) ∧
    ∀vm a c l t.
      num_ics_aux vm a (Cons_monom c l t) =
      a + num_ics_aux vm (num_interp_m vm c l) t) ∧
   ((∀vm. num_interp_cs vm Nil_monom = 0) ∧
    (∀vm l t.
       num_interp_cs vm (Cons_varlist l t) =
       num_ics_aux vm (num_interp_vl vm l) t) ∧
    ∀vm c l t.
      num_interp_cs vm (Cons_monom c l t) =
      num_ics_aux vm (num_interp_m vm c l) t) ∧
   ((∀i. num_spolynom_normalize (SPvar i) = Cons_varlist [i] Nil_monom) ∧
    (∀c. num_spolynom_normalize (SPconst c) = Cons_monom c [] Nil_monom) ∧
    (∀l r.
       num_spolynom_normalize (SPplus l r) =
       num_canonical_sum_merge (num_spolynom_normalize l)
         (num_spolynom_normalize r)) ∧
    ∀l r.
      num_spolynom_normalize (SPmult l r) =
      num_canonical_sum_prod (num_spolynom_normalize l)
        (num_spolynom_normalize r)) ∧
   ∀x.
     num_spolynom_simplify x =
     num_canonical_sum_simplify (num_spolynom_normalize x)

num_rewrites

|- ((∀n. 0 + n = n) ∧ (∀n. n + 0 = n) ∧
    (∀n m. NUMERAL n + NUMERAL m = NUMERAL (numeral$iZ (n + m))) ∧
    (∀n. 0 * n = 0) ∧ (∀n. n * 0 = 0) ∧
    (∀n m. NUMERAL n * NUMERAL m = NUMERAL (n * m)) ∧ (∀n. 0 − n = 0) ∧
    (∀n. n − 0 = n) ∧ (∀n m. NUMERAL n − NUMERAL m = NUMERAL (n − m)) ∧
    (∀n. 0 ** NUMERAL (BIT1 n) = 0) ∧ (∀n. 0 ** NUMERAL (BIT2 n) = 0) ∧
    (∀n. n ** 0 = 1) ∧ (∀n m. NUMERAL n ** NUMERAL m = NUMERAL (n ** m)) ∧
    (SUC 0 = 1) ∧ (∀n. SUC (NUMERAL n) = NUMERAL (SUC n)) ∧ (PRE 0 = 0) ∧
    (∀n. PRE (NUMERAL n) = NUMERAL (PRE n)) ∧
    (∀n. (NUMERAL n = 0) ⇔ (n = ZERO)) ∧ (∀n. (0 = NUMERAL n) ⇔ (n = ZERO)) ∧
    (∀n m. (NUMERAL n = NUMERAL m) ⇔ (n = m)) ∧ (∀n. n < 0 ⇔ F) ∧
    (∀n. 0 < NUMERAL n ⇔ ZERO < n) ∧ (∀n m. NUMERAL n < NUMERAL m ⇔ n < m) ∧
    (∀n. 0 > n ⇔ F) ∧ (∀n. NUMERAL n > 0 ⇔ ZERO < n) ∧
    (∀n m. NUMERAL n > NUMERAL m ⇔ m < n) ∧ (∀n. 0 ≤ n ⇔ T) ∧
    (∀n. NUMERAL n ≤ 0 ⇔ n ≤ ZERO) ∧ (∀n m. NUMERAL n ≤ NUMERAL m ⇔ n ≤ m) ∧
    (∀n. n ≥ 0 ⇔ T) ∧ (∀n. 0 ≥ n ⇔ (n = 0)) ∧
    (∀n m. NUMERAL n ≥ NUMERAL m ⇔ m ≤ n) ∧ (∀n. ODD (NUMERAL n) ⇔ ODD n) ∧
    (∀n. EVEN (NUMERAL n) ⇔ EVEN n) ∧ ¬ODD 0 ∧ EVEN 0) ∧
   (∀n m.
      ((ZERO = BIT1 n) ⇔ F) ∧ ((BIT1 n = ZERO) ⇔ F) ∧ ((ZERO = BIT2 n) ⇔ F) ∧
      ((BIT2 n = ZERO) ⇔ F) ∧ ((BIT1 n = BIT2 m) ⇔ F) ∧
      ((BIT2 n = BIT1 m) ⇔ F) ∧ ((BIT1 n = BIT1 m) ⇔ (n = m)) ∧
      ((BIT2 n = BIT2 m) ⇔ (n = m))) ∧
   ((SUC ZERO = BIT1 ZERO) ∧ (∀n. SUC (BIT1 n) = BIT2 n) ∧
    ∀n. SUC (BIT2 n) = BIT1 (SUC n)) ∧
   ((numeral$iiSUC ZERO = BIT2 ZERO) ∧
    (numeral$iiSUC (BIT1 n) = BIT1 (SUC n)) ∧
    (numeral$iiSUC (BIT2 n) = BIT2 (SUC n))) ∧
   (∀n m.
      (numeral$iZ (ZERO + n) = n) ∧ (numeral$iZ (n + ZERO) = n) ∧
      (numeral$iZ (BIT1 n + BIT1 m) = BIT2 (numeral$iZ (n + m))) ∧
      (numeral$iZ (BIT1 n + BIT2 m) = BIT1 (SUC (n + m))) ∧
      (numeral$iZ (BIT2 n + BIT1 m) = BIT1 (SUC (n + m))) ∧
      (numeral$iZ (BIT2 n + BIT2 m) = BIT2 (SUC (n + m))) ∧
      (SUC (ZERO + n) = SUC n) ∧ (SUC (n + ZERO) = SUC n) ∧
      (SUC (BIT1 n + BIT1 m) = BIT1 (SUC (n + m))) ∧
      (SUC (BIT1 n + BIT2 m) = BIT2 (SUC (n + m))) ∧
      (SUC (BIT2 n + BIT1 m) = BIT2 (SUC (n + m))) ∧
      (SUC (BIT2 n + BIT2 m) = BIT1 (numeral$iiSUC (n + m))) ∧
      (numeral$iiSUC (ZERO + n) = numeral$iiSUC n) ∧
      (numeral$iiSUC (n + ZERO) = numeral$iiSUC n) ∧
      (numeral$iiSUC (BIT1 n + BIT1 m) = BIT2 (SUC (n + m))) ∧
      (numeral$iiSUC (BIT1 n + BIT2 m) = BIT1 (numeral$iiSUC (n + m))) ∧
      (numeral$iiSUC (BIT2 n + BIT1 m) = BIT1 (numeral$iiSUC (n + m))) ∧
      (numeral$iiSUC (BIT2 n + BIT2 m) = BIT2 (numeral$iiSUC (n + m)))) ∧
   (∀n m.
      (ZERO * n = ZERO) ∧ (n * ZERO = ZERO) ∧
      (BIT1 n * m = numeral$iZ (numeral$iDUB (n * m) + m)) ∧
      (BIT2 n * m = numeral$iDUB (numeral$iZ (n * m + m)))) ∧
   (∀n.
      (numeral$iDUB (BIT1 n) = BIT2 (numeral$iDUB n)) ∧
      (numeral$iDUB (BIT2 n) = BIT2 (BIT1 n)) ∧ (numeral$iDUB ZERO = ZERO)) ∧
   ((ZERO = ZERO) ⇔ T) ∧ ((0 = 0) ⇔ T)