Theory "integerRing"

Signature

Definitions

int_interp_p_def

⊢ int_interp_p = interp_p (ring int_0 int_1 $+ $* numeric_negate)

int_polynom_simplify_def

⊢ int_polynom_simplify =
  polynom_simplify (ring int_0 int_1 $+ $* numeric_negate)

int_polynom_normalize_def

⊢ int_polynom_normalize =
  polynom_normalize (ring int_0 int_1 $+ $* numeric_negate)

int_r_canonical_sum_merge_def

⊢ int_r_canonical_sum_merge =
  r_canonical_sum_merge (ring int_0 int_1 $+ $* numeric_negate)

int_r_monom_insert_def

⊢ int_r_monom_insert = r_monom_insert (ring int_0 int_1 $+ $* numeric_negate)

int_r_varlist_insert_def

⊢ int_r_varlist_insert =
  r_varlist_insert (ring int_0 int_1 $+ $* numeric_negate)

int_r_canonical_sum_scalar_def

⊢ int_r_canonical_sum_scalar =
  r_canonical_sum_scalar (ring int_0 int_1 $+ $* numeric_negate)

int_r_canonical_sum_scalar2_def

⊢ int_r_canonical_sum_scalar2 =
  r_canonical_sum_scalar2 (ring int_0 int_1 $+ $* numeric_negate)

int_r_canonical_sum_scalar3_def

⊢ int_r_canonical_sum_scalar3 =
  r_canonical_sum_scalar3 (ring int_0 int_1 $+ $* numeric_negate)

int_r_canonical_sum_prod_def

⊢ int_r_canonical_sum_prod =
  r_canonical_sum_prod (ring int_0 int_1 $+ $* numeric_negate)

int_r_canonical_sum_simplify_def

⊢ int_r_canonical_sum_simplify =
  r_canonical_sum_simplify (ring int_0 int_1 $+ $* numeric_negate)

int_r_ivl_aux_def

⊢ int_r_ivl_aux = r_ivl_aux (ring int_0 int_1 $+ $* numeric_negate)

int_r_interp_vl_def

⊢ int_r_interp_vl = r_interp_vl (ring int_0 int_1 $+ $* numeric_negate)

int_r_interp_m_def

⊢ int_r_interp_m = r_interp_m (ring int_0 int_1 $+ $* numeric_negate)

int_r_ics_aux_def

⊢ int_r_ics_aux = r_ics_aux (ring int_0 int_1 $+ $* numeric_negate)

int_r_interp_cs_def

⊢ int_r_interp_cs = r_interp_cs (ring int_0 int_1 $+ $* numeric_negate)

int_r_spolynom_normalize_def

⊢ int_r_spolynom_normalize =
  r_spolynom_normalize (ring int_0 int_1 $+ $* numeric_negate)

int_r_spolynom_simplify_def

⊢ int_r_spolynom_simplify =
  r_spolynom_simplify (ring int_0 int_1 $+ $* numeric_negate)

int_r_interp_sp_def

⊢ int_r_interp_sp = r_interp_sp (ring int_0 int_1 $+ $* numeric_negate)

Theorems

int_is_ring

⊢ is_ring (ring int_0 int_1 $+ $* numeric_negate)

int_ring_thms

⊢ is_ring (ring int_0 int_1 $+ $* numeric_negate) ∧
  (∀vm p. int_interp_p vm p = int_r_interp_cs vm (int_polynom_simplify p)) ∧
  (((∀vm c. int_interp_p vm (Pconst c) = c) ∧
    (∀vm i. int_interp_p vm (Pvar i) = varmap_find i vm) ∧
    (∀vm p1 p2.
         int_interp_p vm (Pplus p1 p2) =
         int_interp_p vm p1 + int_interp_p vm p2) ∧
    (∀vm p1 p2.
         int_interp_p vm (Pmult p1 p2) =
         int_interp_p vm p1 * int_interp_p vm p2) ∧
    ∀vm p1. int_interp_p vm (Popp p1) = -int_interp_p vm p1) ∧
   (∀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.
        int_r_canonical_sum_merge (Cons_monom c1 l1 t1) (Cons_monom c2 l2 t2) =
        case list_cmp index_compare l1 l2 of
          Less =>
            Cons_monom c1 l1
              (int_r_canonical_sum_merge t1 (Cons_monom c2 l2 t2))
        | Equal => Cons_monom (c1 + c2) l1 (int_r_canonical_sum_merge t1 t2)
        | Greater =>
          Cons_monom c2 l2
            (int_r_canonical_sum_merge (Cons_monom c1 l1 t1) t2)) ∧
   (∀t2 t1 l2 l1 c1.
        int_r_canonical_sum_merge (Cons_monom c1 l1 t1) (Cons_varlist l2 t2) =
        case list_cmp index_compare l1 l2 of
          Less =>
            Cons_monom c1 l1
              (int_r_canonical_sum_merge t1 (Cons_varlist l2 t2))
        | Equal =>
          Cons_monom (c1 + int_1) l1 (int_r_canonical_sum_merge t1 t2)
        | Greater =>
          Cons_varlist l2 (int_r_canonical_sum_merge (Cons_monom c1 l1 t1) t2)) ∧
   (∀t2 t1 l2 l1 c2.
        int_r_canonical_sum_merge (Cons_varlist l1 t1) (Cons_monom c2 l2 t2) =
        case list_cmp index_compare l1 l2 of
          Less =>
            Cons_varlist l1
              (int_r_canonical_sum_merge t1 (Cons_monom c2 l2 t2))
        | Equal =>
          Cons_monom (int_1 + c2) l1 (int_r_canonical_sum_merge t1 t2)
        | Greater =>
          Cons_monom c2 l2 (int_r_canonical_sum_merge (Cons_varlist l1 t1) t2)) ∧
   (∀t2 t1 l2 l1.
        int_r_canonical_sum_merge (Cons_varlist l1 t1) (Cons_varlist l2 t2) =
        case list_cmp index_compare l1 l2 of
          Less =>
            Cons_varlist l1
              (int_r_canonical_sum_merge t1 (Cons_varlist l2 t2))
        | Equal =>
          Cons_monom (int_1 + int_1) l1 (int_r_canonical_sum_merge t1 t2)
        | Greater =>
          Cons_varlist l2 (int_r_canonical_sum_merge (Cons_varlist l1 t1) t2)) ∧
   (∀s1. int_r_canonical_sum_merge s1 Nil_monom = s1) ∧
   (∀v6 v5 v4.
        int_r_canonical_sum_merge Nil_monom (Cons_monom v4 v5 v6) =
        Cons_monom v4 v5 v6) ∧
   ∀v8 v7.
       int_r_canonical_sum_merge Nil_monom (Cons_varlist v7 v8) =
       Cons_varlist v7 v8) ∧
  ((∀t2 l2 l1 c2 c1.
        int_r_monom_insert c1 l1 (Cons_monom c2 l2 t2) =
        case list_cmp index_compare l1 l2 of
          Less => Cons_monom c1 l1 (Cons_monom c2 l2 t2)
        | Equal => Cons_monom (c1 + c2) l1 t2
        | Greater => Cons_monom c2 l2 (int_r_monom_insert c1 l1 t2)) ∧
   (∀t2 l2 l1 c1.
        int_r_monom_insert c1 l1 (Cons_varlist l2 t2) =
        case list_cmp index_compare l1 l2 of
          Less => Cons_monom c1 l1 (Cons_varlist l2 t2)
        | Equal => Cons_monom (c1 + int_1) l1 t2
        | Greater => Cons_varlist l2 (int_r_monom_insert c1 l1 t2)) ∧
   ∀l1 c1. int_r_monom_insert c1 l1 Nil_monom = Cons_monom c1 l1 Nil_monom) ∧
  ((∀t2 l2 l1 c2.
        int_r_varlist_insert l1 (Cons_monom c2 l2 t2) =
        case list_cmp index_compare l1 l2 of
          Less => Cons_varlist l1 (Cons_monom c2 l2 t2)
        | Equal => Cons_monom (int_1 + c2) l1 t2
        | Greater => Cons_monom c2 l2 (int_r_varlist_insert l1 t2)) ∧
   (∀t2 l2 l1.
        int_r_varlist_insert l1 (Cons_varlist l2 t2) =
        case list_cmp index_compare l1 l2 of
          Less => Cons_varlist l1 (Cons_varlist l2 t2)
        | Equal => Cons_monom (int_1 + int_1) l1 t2
        | Greater => Cons_varlist l2 (int_r_varlist_insert l1 t2)) ∧
   ∀l1. int_r_varlist_insert l1 Nil_monom = Cons_varlist l1 Nil_monom) ∧
  ((∀c0 c l t.
        int_r_canonical_sum_scalar c0 (Cons_monom c l t) =
        Cons_monom (c0 * c) l (int_r_canonical_sum_scalar c0 t)) ∧
   (∀c0 l t.
        int_r_canonical_sum_scalar c0 (Cons_varlist l t) =
        Cons_monom c0 l (int_r_canonical_sum_scalar c0 t)) ∧
   ∀c0. int_r_canonical_sum_scalar c0 Nil_monom = Nil_monom) ∧
  ((∀l0 c l t.
        int_r_canonical_sum_scalar2 l0 (Cons_monom c l t) =
        int_r_monom_insert c (list_merge index_lt l0 l)
          (int_r_canonical_sum_scalar2 l0 t)) ∧
   (∀l0 l t.
        int_r_canonical_sum_scalar2 l0 (Cons_varlist l t) =
        int_r_varlist_insert (list_merge index_lt l0 l)
          (int_r_canonical_sum_scalar2 l0 t)) ∧
   ∀l0. int_r_canonical_sum_scalar2 l0 Nil_monom = Nil_monom) ∧
  ((∀c0 l0 c l t.
        int_r_canonical_sum_scalar3 c0 l0 (Cons_monom c l t) =
        int_r_monom_insert (c0 * c) (list_merge index_lt l0 l)
          (int_r_canonical_sum_scalar3 c0 l0 t)) ∧
   (∀c0 l0 l t.
        int_r_canonical_sum_scalar3 c0 l0 (Cons_varlist l t) =
        int_r_monom_insert c0 (list_merge index_lt l0 l)
          (int_r_canonical_sum_scalar3 c0 l0 t)) ∧
   ∀c0 l0. int_r_canonical_sum_scalar3 c0 l0 Nil_monom = Nil_monom) ∧
  ((∀c1 l1 t1 s2.
        int_r_canonical_sum_prod (Cons_monom c1 l1 t1) s2 =
        int_r_canonical_sum_merge (int_r_canonical_sum_scalar3 c1 l1 s2)
          (int_r_canonical_sum_prod t1 s2)) ∧
   (∀l1 t1 s2.
        int_r_canonical_sum_prod (Cons_varlist l1 t1) s2 =
        int_r_canonical_sum_merge (int_r_canonical_sum_scalar2 l1 s2)
          (int_r_canonical_sum_prod t1 s2)) ∧
   ∀s2. int_r_canonical_sum_prod Nil_monom s2 = Nil_monom) ∧
  ((∀c l t.
        int_r_canonical_sum_simplify (Cons_monom c l t) =
        if c = int_0 then int_r_canonical_sum_simplify t
        else if c = int_1 then Cons_varlist l (int_r_canonical_sum_simplify t)
        else Cons_monom c l (int_r_canonical_sum_simplify t)) ∧
   (∀l t.
        int_r_canonical_sum_simplify (Cons_varlist l t) =
        Cons_varlist l (int_r_canonical_sum_simplify t)) ∧
   int_r_canonical_sum_simplify Nil_monom = Nil_monom) ∧
  ((∀vm x. int_r_ivl_aux vm x [] = varmap_find x vm) ∧
   ∀vm x x' t'.
       int_r_ivl_aux vm x (x'::t') = varmap_find x vm * int_r_ivl_aux vm x' t') ∧
  ((∀vm. int_r_interp_vl vm [] = int_1) ∧
   ∀vm x t. int_r_interp_vl vm (x::t) = int_r_ivl_aux vm x t) ∧
  ((∀vm c. int_r_interp_m vm c [] = c) ∧
   ∀vm c x t. int_r_interp_m vm c (x::t) = c * int_r_ivl_aux vm x t) ∧
  ((∀vm a. int_r_ics_aux vm a Nil_monom = a) ∧
   (∀vm a l t.
        int_r_ics_aux vm a (Cons_varlist l t) =
        a + int_r_ics_aux vm (int_r_interp_vl vm l) t) ∧
   ∀vm a c l t.
       int_r_ics_aux vm a (Cons_monom c l t) =
       a + int_r_ics_aux vm (int_r_interp_m vm c l) t) ∧
  ((∀vm. int_r_interp_cs vm Nil_monom = int_0) ∧
   (∀vm l t.
        int_r_interp_cs vm (Cons_varlist l t) =
        int_r_ics_aux vm (int_r_interp_vl vm l) t) ∧
   ∀vm c l t.
       int_r_interp_cs vm (Cons_monom c l t) =
       int_r_ics_aux vm (int_r_interp_m vm c l) t) ∧
  ((∀i. int_polynom_normalize (Pvar i) = Cons_varlist [i] Nil_monom) ∧
   (∀c. int_polynom_normalize (Pconst c) = Cons_monom c [] Nil_monom) ∧
   (∀pl pr.
        int_polynom_normalize (Pplus pl pr) =
        int_r_canonical_sum_merge (int_polynom_normalize pl)
          (int_polynom_normalize pr)) ∧
   (∀pl pr.
        int_polynom_normalize (Pmult pl pr) =
        int_r_canonical_sum_prod (int_polynom_normalize pl)
          (int_polynom_normalize pr)) ∧
   ∀p.
       int_polynom_normalize (Popp p) =
       int_r_canonical_sum_scalar3 (-int_1) [] (int_polynom_normalize p)) ∧
  ∀x.
      int_polynom_simplify x =
      int_r_canonical_sum_simplify (int_polynom_normalize x)

int_calculate

⊢ &n + &m = &(n + m) ∧ -&n + &m = (if n ≤ m then &(m − n) else -&(n − m)) ∧
  &n + -&m = (if m ≤ n then &(n − m) else -&(m − n)) ∧ -&n + -&m = -&(n + m) ∧
  &n * &m = &(n * m) ∧ -&n * &m = -&(n * m) ∧ &n * -&m = -&(n * m) ∧
  -&n * -&m = &(n * m) ∧ (&n = &m ⇔ n = m) ∧ (&n = -&m ⇔ n = 0 ∧ m = 0) ∧
  (-&n = &m ⇔ n = 0 ∧ m = 0) ∧ (-&n = -&m ⇔ n = m) ∧ - -x = x ∧ -0 = 0

int_rewrites

⊢ (&n + &m = &(n + m) ∧ -&n + &m = (if n ≤ m then &(m − n) else -&(n − m)) ∧
   &n + -&m = (if m ≤ n then &(n − m) else -&(m − n)) ∧
   -&n + -&m = -&(n + m) ∧ &n * &m = &(n * m) ∧ -&n * &m = -&(n * m) ∧
   &n * -&m = -&(n * m) ∧ -&n * -&m = &(n * m) ∧ (&n = &m ⇔ n = m) ∧
   (&n = -&m ⇔ n = 0 ∧ m = 0) ∧ (-&n = &m ⇔ n = 0 ∧ m = 0) ∧
   (-&n = -&m ⇔ n = m) ∧ - -x = x ∧ -0 = 0) ∧ int_0 = 0 ∧ int_1 = 1 ∧
  (∀n m.
       (ZERO < BIT1 n ⇔ T) ∧ (ZERO < BIT2 n ⇔ T) ∧ (n < ZERO ⇔ F) ∧
       (BIT1 n < BIT1 m ⇔ n < m) ∧ (BIT2 n < BIT2 m ⇔ n < m) ∧
       (BIT1 n < BIT2 m ⇔ ¬(m < n)) ∧ (BIT2 n < BIT1 m ⇔ n < m)) ∧
  (∀n m.
       (ZERO ≤ n ⇔ T) ∧ (BIT1 n ≤ ZERO ⇔ F) ∧ (BIT2 n ≤ ZERO ⇔ F) ∧
       (BIT1 n ≤ BIT1 m ⇔ n ≤ m) ∧ (BIT1 n ≤ BIT2 m ⇔ n ≤ m) ∧
       (BIT2 n ≤ BIT1 m ⇔ ¬(m ≤ n)) ∧ (BIT2 n ≤ BIT2 m ⇔ n ≤ m)) ∧
  (∀n m. NUMERAL (n − m) = if m < n then NUMERAL (numeral$iSUB T n m) else 0) ∧
  (∀b n m.
       numeral$iSUB b ZERO x = ZERO ∧ numeral$iSUB T n ZERO = n ∧
       numeral$iSUB F (BIT1 n) ZERO = numeral$iDUB n ∧
       numeral$iSUB T (BIT1 n) (BIT1 m) = numeral$iDUB (numeral$iSUB T n m) ∧
       numeral$iSUB F (BIT1 n) (BIT1 m) = BIT1 (numeral$iSUB F n m) ∧
       numeral$iSUB T (BIT1 n) (BIT2 m) = BIT1 (numeral$iSUB F n m) ∧
       numeral$iSUB F (BIT1 n) (BIT2 m) = numeral$iDUB (numeral$iSUB F n m) ∧
       numeral$iSUB F (BIT2 n) ZERO = BIT1 n ∧
       numeral$iSUB T (BIT2 n) (BIT1 m) = BIT1 (numeral$iSUB T n m) ∧
       numeral$iSUB F (BIT2 n) (BIT1 m) = numeral$iDUB (numeral$iSUB T n m) ∧
       numeral$iSUB T (BIT2 n) (BIT2 m) = numeral$iDUB (numeral$iSUB T n m) ∧
       numeral$iSUB F (BIT2 n) (BIT2 m) = BIT1 (numeral$iSUB F n m)) ∧
  ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)