Theory: integerRing

Parents

Types

Constants

int_r_ivl_aux :int varmap -> index -> index list -> int
int_r_interp_cs :int varmap -> int canonical_sum -> int
int_r_interp_sp :int varmap -> int spolynom -> int
int_r_interp_vl :int varmap -> index list -> int
int_r_spolynom_normalize :int spolynom -> int canonical_sum
int_r_varlist_insert :index list -> int canonical_sum -> int canonical_sum
int_r_ics_aux :int varmap -> int -> int canonical_sum -> int
int_interp_p :int varmap -> int polynom -> int
int_r_interp_m :int varmap -> int -> index list -> int
int_polynom_normalize :int polynom -> int canonical_sum
int_r_canonical_sum_merge :int canonical_sum -> int canonical_sum -> int canonical_sum
int_polynom_simplify :int polynom -> int canonical_sum
int_r_canonical_sum_scalar :int -> int canonical_sum -> int canonical_sum
int_r_canonical_sum_prod :int canonical_sum -> int canonical_sum -> int canonical_sum
int_r_canonical_sum_simplify :int canonical_sum -> int canonical_sum
int_r_canonical_sum_scalar2 :index list -> int canonical_sum -> int canonical_sum
int_r_canonical_sum_scalar3 :int -> index list -> int canonical_sum -> int canonical_sum
int_r_spolynom_simplify :int spolynom -> int canonical_sum
int_r_monom_insert :int -> index list -> int canonical_sum -> int canonical_sum

Definitions

int_interp_p_def

|- int_interp_p = interp_p (ring int_0 int_1 $+ $* $~)

int_polynom_normalize_def

|- int_polynom_normalize = polynom_normalize (ring int_0 int_1 $+ $* $~)

int_polynom_simplify_def

|- int_polynom_simplify = polynom_simplify (ring int_0 int_1 $+ $* $~)

int_r_canonical_sum_merge_def

|- int_r_canonical_sum_merge =
   r_canonical_sum_merge (ring int_0 int_1 $+ $* $~)

int_r_canonical_sum_prod_def

|- int_r_canonical_sum_prod = r_canonical_sum_prod (ring int_0 int_1 $+ $* $~)

int_r_canonical_sum_scalar2_def

|- int_r_canonical_sum_scalar2 =
   r_canonical_sum_scalar2 (ring int_0 int_1 $+ $* $~)

int_r_canonical_sum_scalar3_def

|- int_r_canonical_sum_scalar3 =
   r_canonical_sum_scalar3 (ring int_0 int_1 $+ $* $~)

int_r_canonical_sum_scalar_def

|- int_r_canonical_sum_scalar =
   r_canonical_sum_scalar (ring int_0 int_1 $+ $* $~)

int_r_canonical_sum_simplify_def

|- int_r_canonical_sum_simplify =
   r_canonical_sum_simplify (ring int_0 int_1 $+ $* $~)

int_r_ics_aux_def

|- int_r_ics_aux = r_ics_aux (ring int_0 int_1 $+ $* $~)

int_r_interp_cs_def

|- int_r_interp_cs = r_interp_cs (ring int_0 int_1 $+ $* $~)

int_r_interp_m_def

|- int_r_interp_m = r_interp_m (ring int_0 int_1 $+ $* $~)

int_r_interp_sp_def

|- int_r_interp_sp = r_interp_sp (ring int_0 int_1 $+ $* $~)

int_r_interp_vl_def

|- int_r_interp_vl = r_interp_vl (ring int_0 int_1 $+ $* $~)

int_r_ivl_aux_def

|- int_r_ivl_aux = r_ivl_aux (ring int_0 int_1 $+ $* $~)

int_r_monom_insert_def

|- int_r_monom_insert = r_monom_insert (ring int_0 int_1 $+ $* $~)

int_r_spolynom_normalize_def

|- int_r_spolynom_normalize = r_spolynom_normalize (ring int_0 int_1 $+ $* $~)

int_r_spolynom_simplify_def

|- int_r_spolynom_simplify = r_spolynom_simplify (ring int_0 int_1 $+ $* $~)

int_r_varlist_insert_def

|- int_r_varlist_insert = r_varlist_insert (ring int_0 int_1 $+ $* $~)

Theorems

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_is_ring

|- is_ring (ring int_0 int_1 $+ $* $~)

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.
      (ALT_ZERO < NUMERAL_BIT1 n = T) /\ (ALT_ZERO < NUMERAL_BIT2 n = T) /\
      (n < ALT_ZERO = F) /\ (NUMERAL_BIT1 n < NUMERAL_BIT1 m = n < m) /\
      (NUMERAL_BIT2 n < NUMERAL_BIT2 m = n < m) /\
      (NUMERAL_BIT1 n < NUMERAL_BIT2 m = ~(m < n)) /\
      (NUMERAL_BIT2 n < NUMERAL_BIT1 m = n < m)) /\
   (!n m.
      (ALT_ZERO <= n = T) /\ (NUMERAL_BIT1 n <= ALT_ZERO = F) /\
      (NUMERAL_BIT2 n <= ALT_ZERO = F) /\
      (NUMERAL_BIT1 n <= NUMERAL_BIT1 m = n <= m) /\
      (NUMERAL_BIT1 n <= NUMERAL_BIT2 m = n <= m) /\
      (NUMERAL_BIT2 n <= NUMERAL_BIT1 m = ~(m <= n)) /\
      (NUMERAL_BIT2 n <= NUMERAL_BIT2 m = n <= m)) /\
   (!n m. NUMERAL (n - m) = (if m < n then NUMERAL (iSUB T n m) else 0)) /\
   (!b n m.
      (iSUB b ALT_ZERO x = ALT_ZERO) /\ (iSUB T n ALT_ZERO = n) /\
      (iSUB F (NUMERAL_BIT1 n) ALT_ZERO = iDUB n) /\
      (iSUB T (NUMERAL_BIT1 n) (NUMERAL_BIT1 m) = iDUB (iSUB T n m)) /\
      (iSUB F (NUMERAL_BIT1 n) (NUMERAL_BIT1 m) =
       NUMERAL_BIT1 (iSUB F n m)) /\
      (iSUB T (NUMERAL_BIT1 n) (NUMERAL_BIT2 m) =
       NUMERAL_BIT1 (iSUB F n m)) /\
      (iSUB F (NUMERAL_BIT1 n) (NUMERAL_BIT2 m) = iDUB (iSUB F n m)) /\
      (iSUB F (NUMERAL_BIT2 n) ALT_ZERO = NUMERAL_BIT1 n) /\
      (iSUB T (NUMERAL_BIT2 n) (NUMERAL_BIT1 m) =
       NUMERAL_BIT1 (iSUB T n m)) /\
      (iSUB F (NUMERAL_BIT2 n) (NUMERAL_BIT1 m) = iDUB (iSUB T n m)) /\
      (iSUB T (NUMERAL_BIT2 n) (NUMERAL_BIT2 m) = iDUB (iSUB T n m)) /\
      (iSUB F (NUMERAL_BIT2 n) (NUMERAL_BIT2 m) =
       NUMERAL_BIT1 (iSUB F n m))) /\
   !t.
     (T /\ t = t) /\ (t /\ T = t) /\ (F /\ t = F) /\ (t /\ F = F) /\
     (t /\ t = t)

int_ring_thms

|- is_ring (ring int_0 int_1 $+ $* $~) /\
   (!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) /\
    (varmap_find End_idx (Node_vm x v1 v2) = x) /\
    (varmap_find (Right_idx i1) (Node_vm x v1 v2) = varmap_find i1 v2) /\
    (varmap_find (Left_idx i1) (Node_vm x v1 v2) = varmap_find i1 v1) /\
    (varmap_find End_idx Empty_vm = @x. T) /\
    (varmap_find (Right_idx v5) Empty_vm = @x. T) /\
    (varmap_find (Left_idx v4) Empty_vm = @x. T)) /\
   ((int_r_canonical_sum_merge (Cons_monom c1 l1 t1) (Cons_monom c2 l2 t2) =
     compare (list_compare index_compare l1 l2)
       (Cons_monom c1 l1 (int_r_canonical_sum_merge t1 (Cons_monom c2 l2 t2)))
       (Cons_monom (c1 + c2) l1 (int_r_canonical_sum_merge t1 t2))
       (Cons_monom c2 l2
          (int_r_canonical_sum_merge (Cons_monom c1 l1 t1) t2))) /\
    (int_r_canonical_sum_merge (Cons_monom c1 l1 t1) (Cons_varlist l2 t2) =
     compare (list_compare index_compare l1 l2)
       (Cons_monom c1 l1 (int_r_canonical_sum_merge t1 (Cons_varlist l2 t2)))
       (Cons_monom (c1 + int_1) l1 (int_r_canonical_sum_merge t1 t2))
       (Cons_varlist l2
          (int_r_canonical_sum_merge (Cons_monom c1 l1 t1) t2))) /\
    (int_r_canonical_sum_merge (Cons_varlist l1 t1) (Cons_monom c2 l2 t2) =
     compare (list_compare index_compare l1 l2)
       (Cons_varlist l1 (int_r_canonical_sum_merge t1 (Cons_monom c2 l2 t2)))
       (Cons_monom (int_1 + c2) l1 (int_r_canonical_sum_merge t1 t2))
       (Cons_monom c2 l2
          (int_r_canonical_sum_merge (Cons_varlist l1 t1) t2))) /\
    (int_r_canonical_sum_merge (Cons_varlist l1 t1) (Cons_varlist l2 t2) =
     compare (list_compare index_compare l1 l2)
       (Cons_varlist l1 (int_r_canonical_sum_merge t1 (Cons_varlist l2 t2)))
       (Cons_monom (int_1 + int_1) l1 (int_r_canonical_sum_merge t1 t2))
       (Cons_varlist l2
          (int_r_canonical_sum_merge (Cons_varlist l1 t1) t2))) /\
    (int_r_canonical_sum_merge (Cons_varlist v7 v8) Nil_monom =
     Cons_varlist v7 v8) /\
    (int_r_canonical_sum_merge (Cons_monom v4 v5 v6) Nil_monom =
     Cons_monom v4 v5 v6) /\
    (int_r_canonical_sum_merge Nil_monom Nil_monom = Nil_monom) /\
    (int_r_canonical_sum_merge Nil_monom (Cons_varlist v17 v18) =
     Cons_varlist v17 v18) /\
    (int_r_canonical_sum_merge Nil_monom (Cons_monom v14 v15 v16) =
     Cons_monom v14 v15 v16)) /\
   ((int_r_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 (int_r_monom_insert c1 l1 t2))) /\
    (int_r_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 + int_1) l1 t2)
       (Cons_varlist l2 (int_r_monom_insert c1 l1 t2))) /\
    (int_r_monom_insert c1 l1 Nil_monom = Cons_monom c1 l1 Nil_monom)) /\
   ((int_r_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 (int_1 + c2) l1 t2)
       (Cons_monom c2 l2 (int_r_varlist_insert l1 t2))) /\
    (int_r_varlist_insert l1 (Cons_varlist l2 t2) =
     compare (list_compare index_compare l1 l2)
       (Cons_varlist l1 (Cons_varlist l2 t2))
       (Cons_monom (int_1 + int_1) l1 t2)
       (Cons_varlist l2 (int_r_varlist_insert l1 t2))) /\
    (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)