Theory "ringNorm"

Signature

Definitions

r_interp_sp_def

|- ∀r. r_interp_sp r = interp_sp (semi_ring_of r)

r_spolynom_simplify_def

|- ∀r. r_spolynom_simplify r = spolynom_simplify (semi_ring_of r)

r_spolynom_normalize_def

|- ∀r. r_spolynom_normalize r = spolynom_normalize (semi_ring_of r)

r_interp_cs_def

|- ∀r. r_interp_cs r = interp_cs (semi_ring_of r)

r_ics_aux_def

|- ∀r. r_ics_aux r = ics_aux (semi_ring_of r)

r_interp_m_def

|- ∀r. r_interp_m r = interp_m (semi_ring_of r)

r_interp_vl_def

|- ∀r. r_interp_vl r = interp_vl (semi_ring_of r)

r_ivl_aux_def

|- ∀r. r_ivl_aux r = ivl_aux (semi_ring_of r)

r_canonical_sum_simplify_def

|- ∀r. r_canonical_sum_simplify r = canonical_sum_simplify (semi_ring_of r)

r_canonical_sum_prod_def

|- ∀r. r_canonical_sum_prod r = canonical_sum_prod (semi_ring_of r)

r_canonical_sum_scalar3_def

|- ∀r. r_canonical_sum_scalar3 r = canonical_sum_scalar3 (semi_ring_of r)

r_canonical_sum_scalar2_def

|- ∀r. r_canonical_sum_scalar2 r = canonical_sum_scalar2 (semi_ring_of r)

r_canonical_sum_scalar_def

|- ∀r. r_canonical_sum_scalar r = canonical_sum_scalar (semi_ring_of r)

r_varlist_insert_def

|- ∀r. r_varlist_insert r = varlist_insert (semi_ring_of r)

r_monom_insert_def

|- ∀r. r_monom_insert r = monom_insert (semi_ring_of r)

r_canonical_sum_merge_def

|- ∀r. r_canonical_sum_merge r = canonical_sum_merge (semi_ring_of r)

polynom_TY_DEF

|- ∃rep.
     TYPE_DEFINITION
       (λa0'.
          ∀'polynom' .
            (∀a0'.
               (∃a.
                  a0' =
                  (λa. ind_type$CONSTR 0 (a,ARB) (λn. ind_type$BOTTOM)) a) ∨
               (∃a.
                  a0' =
                  (λa. ind_type$CONSTR (SUC 0) (ARB,a) (λn. ind_type$BOTTOM))
                    a) ∨
               (∃a0 a1.
                  (a0' =
                   (λa0 a1.
                      ind_type$CONSTR (SUC (SUC 0)) (ARB,ARB)
                        (ind_type$FCONS a0
                           (ind_type$FCONS a1 (λn. ind_type$BOTTOM)))) a0
                     a1) ∧ 'polynom' a0 ∧ 'polynom' a1) ∨
               (∃a0 a1.
                  (a0' =
                   (λa0 a1.
                      ind_type$CONSTR (SUC (SUC (SUC 0))) (ARB,ARB)
                        (ind_type$FCONS a0
                           (ind_type$FCONS a1 (λn. ind_type$BOTTOM)))) a0
                     a1) ∧ 'polynom' a0 ∧ 'polynom' a1) ∨
               (∃a.
                  (a0' =
                   (λa.
                      ind_type$CONSTR (SUC (SUC (SUC (SUC 0)))) (ARB,ARB)
                        (ind_type$FCONS a (λn. ind_type$BOTTOM))) a) ∧
                  'polynom' a) ⇒
               'polynom' a0') ⇒
            'polynom' a0') rep

polynom_case_def

|- (∀a f f1 f2 f3 f4. polynom_CASE (Pvar a) f f1 f2 f3 f4 = f a) ∧
   (∀a f f1 f2 f3 f4. polynom_CASE (Pconst a) f f1 f2 f3 f4 = f1 a) ∧
   (∀a0 a1 f f1 f2 f3 f4.
      polynom_CASE (Pplus a0 a1) f f1 f2 f3 f4 = f2 a0 a1) ∧
   (∀a0 a1 f f1 f2 f3 f4.
      polynom_CASE (Pmult a0 a1) f f1 f2 f3 f4 = f3 a0 a1) ∧
   ∀a f f1 f2 f3 f4. polynom_CASE (Popp a) f f1 f2 f3 f4 = f4 a

polynom_size_def

|- (∀f a. polynom_size f (Pvar a) = 1 + index_size a) ∧
   (∀f a. polynom_size f (Pconst a) = 1 + f a) ∧
   (∀f a0 a1.
      polynom_size f (Pplus a0 a1) =
      1 + (polynom_size f a0 + polynom_size f a1)) ∧
   (∀f a0 a1.
      polynom_size f (Pmult a0 a1) =
      1 + (polynom_size f a0 + polynom_size f a1)) ∧
   ∀f a. polynom_size f (Popp a) = 1 + polynom_size f a

polynom_normalize_def

|- (∀r i. polynom_normalize r (Pvar i) = Cons_varlist [i] Nil_monom) ∧
   (∀r c. polynom_normalize r (Pconst c) = Cons_monom c [] Nil_monom) ∧
   (∀r pl pr.
      polynom_normalize r (Pplus pl pr) =
      r_canonical_sum_merge r (polynom_normalize r pl)
        (polynom_normalize r pr)) ∧
   (∀r pl pr.
      polynom_normalize r (Pmult pl pr) =
      r_canonical_sum_prod r (polynom_normalize r pl)
        (polynom_normalize r pr)) ∧
   ∀r p.
     polynom_normalize r (Popp p) =
     r_canonical_sum_scalar3 r (RN r (R1 r)) [] (polynom_normalize r p)

polynom_simplify_def

|- ∀r x.
     polynom_simplify r x = r_canonical_sum_simplify r (polynom_normalize r x)

interp_p_def

|- (∀r vm c. interp_p r vm (Pconst c) = c) ∧
   (∀r vm i. interp_p r vm (Pvar i) = varmap_find i vm) ∧
   (∀r vm p1 p2.
      interp_p r vm (Pplus p1 p2) =
      RP r (interp_p r vm p1) (interp_p r vm p2)) ∧
   (∀r vm p1 p2.
      interp_p r vm (Pmult p1 p2) =
      RM r (interp_p r vm p1) (interp_p r vm p2)) ∧
   ∀r vm p1. interp_p r vm (Popp p1) = RN r (interp_p r vm p1)

Theorems

interp_sp_def

|- ∀r.
     (∀vm c. r_interp_sp r vm (SPconst c) = c) ∧
     (∀vm i. r_interp_sp r vm (SPvar i) = varmap_find i vm) ∧
     (∀vm p1 p2.
        r_interp_sp r vm (SPplus p1 p2) =
        RP r (r_interp_sp r vm p1) (r_interp_sp r vm p2)) ∧
     ∀vm p1 p2.
       r_interp_sp r vm (SPmult p1 p2) =
       RM r (r_interp_sp r vm p1) (r_interp_sp r vm p2)

canonical_sum_merge_def

|- ∀r.
     (∀t2 t1 l2 l1 c2 c1.
        r_canonical_sum_merge r (Cons_monom c1 l1 t1) (Cons_monom c2 l2 t2) =
        compare (list_compare index_compare l1 l2)
          (Cons_monom c1 l1
             (r_canonical_sum_merge r t1 (Cons_monom c2 l2 t2)))
          (Cons_monom (RP r c1 c2) l1 (r_canonical_sum_merge r t1 t2))
          (Cons_monom c2 l2
             (r_canonical_sum_merge r (Cons_monom c1 l1 t1) t2))) ∧
     (∀t2 t1 l2 l1 c1.
        r_canonical_sum_merge r (Cons_monom c1 l1 t1) (Cons_varlist l2 t2) =
        compare (list_compare index_compare l1 l2)
          (Cons_monom c1 l1 (r_canonical_sum_merge r t1 (Cons_varlist l2 t2)))
          (Cons_monom (RP r c1 (R1 r)) l1 (r_canonical_sum_merge r t1 t2))
          (Cons_varlist l2
             (r_canonical_sum_merge r (Cons_monom c1 l1 t1) t2))) ∧
     (∀t2 t1 l2 l1 c2.
        r_canonical_sum_merge r (Cons_varlist l1 t1) (Cons_monom c2 l2 t2) =
        compare (list_compare index_compare l1 l2)
          (Cons_varlist l1 (r_canonical_sum_merge r t1 (Cons_monom c2 l2 t2)))
          (Cons_monom (RP r (R1 r) c2) l1 (r_canonical_sum_merge r t1 t2))
          (Cons_monom c2 l2
             (r_canonical_sum_merge r (Cons_varlist l1 t1) t2))) ∧
     (∀t2 t1 l2 l1.
        r_canonical_sum_merge r (Cons_varlist l1 t1) (Cons_varlist l2 t2) =
        compare (list_compare index_compare l1 l2)
          (Cons_varlist l1 (r_canonical_sum_merge r t1 (Cons_varlist l2 t2)))
          (Cons_monom (RP r (R1 r) (R1 r)) l1 (r_canonical_sum_merge r t1 t2))
          (Cons_varlist l2
             (r_canonical_sum_merge r (Cons_varlist l1 t1) t2))) ∧
     (∀s1. r_canonical_sum_merge r s1 Nil_monom = s1) ∧
     (∀v6 v5 v4.
        r_canonical_sum_merge r Nil_monom (Cons_monom v4 v5 v6) =
        Cons_monom v4 v5 v6) ∧
     ∀v8 v7.
       r_canonical_sum_merge r Nil_monom (Cons_varlist v7 v8) =
       Cons_varlist v7 v8

monom_insert_def

|- ∀r.
     (∀t2 l2 l1 c2 c1.
        r_monom_insert r 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 (RP r c1 c2) l1 t2)
          (Cons_monom c2 l2 (r_monom_insert r c1 l1 t2))) ∧
     (∀t2 l2 l1 c1.
        r_monom_insert r c1 l1 (Cons_varlist l2 t2) =
        compare (list_compare index_compare l1 l2)
          (Cons_monom c1 l1 (Cons_varlist l2 t2))
          (Cons_monom (RP r c1 (R1 r)) l1 t2)
          (Cons_varlist l2 (r_monom_insert r c1 l1 t2))) ∧
     ∀l1 c1. r_monom_insert r c1 l1 Nil_monom = Cons_monom c1 l1 Nil_monom

varlist_insert_def

|- ∀r.
     (∀t2 l2 l1 c2.
        r_varlist_insert r l1 (Cons_monom c2 l2 t2) =
        compare (list_compare index_compare l1 l2)
          (Cons_varlist l1 (Cons_monom c2 l2 t2))
          (Cons_monom (RP r (R1 r) c2) l1 t2)
          (Cons_monom c2 l2 (r_varlist_insert r l1 t2))) ∧
     (∀t2 l2 l1.
        r_varlist_insert r l1 (Cons_varlist l2 t2) =
        compare (list_compare index_compare l1 l2)
          (Cons_varlist l1 (Cons_varlist l2 t2))
          (Cons_monom (RP r (R1 r) (R1 r)) l1 t2)
          (Cons_varlist l2 (r_varlist_insert r l1 t2))) ∧
     ∀l1. r_varlist_insert r l1 Nil_monom = Cons_varlist l1 Nil_monom

canonical_sum_scalar_def

|- ∀r.
     (∀c0 c l t.
        r_canonical_sum_scalar r c0 (Cons_monom c l t) =
        Cons_monom (RM r c0 c) l (r_canonical_sum_scalar r c0 t)) ∧
     (∀c0 l t.
        r_canonical_sum_scalar r c0 (Cons_varlist l t) =
        Cons_monom c0 l (r_canonical_sum_scalar r c0 t)) ∧
     ∀c0. r_canonical_sum_scalar r c0 Nil_monom = Nil_monom

canonical_sum_scalar2_def

|- ∀r.
     (∀l0 c l t.
        r_canonical_sum_scalar2 r l0 (Cons_monom c l t) =
        r_monom_insert r c (list_merge index_lt l0 l)
          (r_canonical_sum_scalar2 r l0 t)) ∧
     (∀l0 l t.
        r_canonical_sum_scalar2 r l0 (Cons_varlist l t) =
        r_varlist_insert r (list_merge index_lt l0 l)
          (r_canonical_sum_scalar2 r l0 t)) ∧
     ∀l0. r_canonical_sum_scalar2 r l0 Nil_monom = Nil_monom

canonical_sum_scalar3_def

|- ∀r.
     (∀c0 l0 c l t.
        r_canonical_sum_scalar3 r c0 l0 (Cons_monom c l t) =
        r_monom_insert r (RM r c0 c) (list_merge index_lt l0 l)
          (r_canonical_sum_scalar3 r c0 l0 t)) ∧
     (∀c0 l0 l t.
        r_canonical_sum_scalar3 r c0 l0 (Cons_varlist l t) =
        r_monom_insert r c0 (list_merge index_lt l0 l)
          (r_canonical_sum_scalar3 r c0 l0 t)) ∧
     ∀c0 l0. r_canonical_sum_scalar3 r c0 l0 Nil_monom = Nil_monom

canonical_sum_prod_def

|- ∀r.
     (∀c1 l1 t1 s2.
        r_canonical_sum_prod r (Cons_monom c1 l1 t1) s2 =
        r_canonical_sum_merge r (r_canonical_sum_scalar3 r c1 l1 s2)
          (r_canonical_sum_prod r t1 s2)) ∧
     (∀l1 t1 s2.
        r_canonical_sum_prod r (Cons_varlist l1 t1) s2 =
        r_canonical_sum_merge r (r_canonical_sum_scalar2 r l1 s2)
          (r_canonical_sum_prod r t1 s2)) ∧
     ∀s2. r_canonical_sum_prod r Nil_monom s2 = Nil_monom

canonical_sum_simplify_def

|- ∀r.
     (∀c l t.
        r_canonical_sum_simplify r (Cons_monom c l t) =
        if c = R0 r then r_canonical_sum_simplify r t
        else if c = R1 r then Cons_varlist l (r_canonical_sum_simplify r t)
        else Cons_monom c l (r_canonical_sum_simplify r t)) ∧
     (∀l t.
        r_canonical_sum_simplify r (Cons_varlist l t) =
        Cons_varlist l (r_canonical_sum_simplify r t)) ∧
     (r_canonical_sum_simplify r Nil_monom = Nil_monom)

ivl_aux_def

|- ∀r.
     (∀vm x. r_ivl_aux r vm x [] = varmap_find x vm) ∧
     ∀vm x x' t'.
       r_ivl_aux r vm x (x'::t') =
       RM r (varmap_find x vm) (r_ivl_aux r vm x' t')

interp_vl_def

|- ∀r.
     (∀vm. r_interp_vl r vm [] = R1 r) ∧
     ∀vm x t. r_interp_vl r vm (x::t) = r_ivl_aux r vm x t

interp_m_def

|- ∀r.
     (∀vm c. r_interp_m r vm c [] = c) ∧
     ∀vm c x t. r_interp_m r vm c (x::t) = RM r c (r_ivl_aux r vm x t)

ics_aux_def

|- ∀r.
     (∀vm a. r_ics_aux r vm a Nil_monom = a) ∧
     (∀vm a l t.
        r_ics_aux r vm a (Cons_varlist l t) =
        RP r a (r_ics_aux r vm (r_interp_vl r vm l) t)) ∧
     ∀vm a c l t.
       r_ics_aux r vm a (Cons_monom c l t) =
       RP r a (r_ics_aux r vm (r_interp_m r vm c l) t)

interp_cs_def

|- ∀r.
     (∀vm. r_interp_cs r vm Nil_monom = R0 r) ∧
     (∀vm l t.
        r_interp_cs r vm (Cons_varlist l t) =
        r_ics_aux r vm (r_interp_vl r vm l) t) ∧
     ∀vm c l t.
       r_interp_cs r vm (Cons_monom c l t) =
       r_ics_aux r vm (r_interp_m r vm c l) t

spolynom_normalize_def

|- ∀r.
     (∀i. r_spolynom_normalize r (SPvar i) = Cons_varlist [i] Nil_monom) ∧
     (∀c. r_spolynom_normalize r (SPconst c) = Cons_monom c [] Nil_monom) ∧
     (∀l r'.
        r_spolynom_normalize r (SPplus l r') =
        r_canonical_sum_merge r (r_spolynom_normalize r l)
          (r_spolynom_normalize r r')) ∧
     ∀l r'.
       r_spolynom_normalize r (SPmult l r') =
       r_canonical_sum_prod r (r_spolynom_normalize r l)
         (r_spolynom_normalize r r')

spolynom_simplify_def

|- ∀r x.
     r_spolynom_simplify r x =
     r_canonical_sum_simplify r (r_spolynom_normalize r x)

datatype_polynom

|- DATATYPE (polynom Pvar Pconst Pplus Pmult Popp)

polynom_11

|- (∀a a'. (Pvar a = Pvar a') ⇔ (a = a')) ∧
   (∀a a'. (Pconst a = Pconst a') ⇔ (a = a')) ∧
   (∀a0 a1 a0' a1'. (Pplus a0 a1 = Pplus a0' a1') ⇔ (a0 = a0') ∧ (a1 = a1')) ∧
   (∀a0 a1 a0' a1'. (Pmult a0 a1 = Pmult a0' a1') ⇔ (a0 = a0') ∧ (a1 = a1')) ∧
   ∀a a'. (Popp a = Popp a') ⇔ (a = a')

polynom_distinct

|- (∀a' a. Pvar a ≠ Pconst a') ∧ (∀a1 a0 a. Pvar a ≠ Pplus a0 a1) ∧
   (∀a1 a0 a. Pvar a ≠ Pmult a0 a1) ∧ (∀a' a. Pvar a ≠ Popp a') ∧
   (∀a1 a0 a. Pconst a ≠ Pplus a0 a1) ∧ (∀a1 a0 a. Pconst a ≠ Pmult a0 a1) ∧
   (∀a' a. Pconst a ≠ Popp a') ∧
   (∀a1' a1 a0' a0. Pplus a0 a1 ≠ Pmult a0' a1') ∧
   (∀a1 a0 a. Pplus a0 a1 ≠ Popp a) ∧ ∀a1 a0 a. Pmult a0 a1 ≠ Popp a

polynom_case_cong

|- ∀M M' f f1 f2 f3 f4.
     (M = M') ∧ (∀a. (M' = Pvar a) ⇒ (f a = f' a)) ∧
     (∀a. (M' = Pconst a) ⇒ (f1 a = f1' a)) ∧
     (∀a0 a1. (M' = Pplus a0 a1) ⇒ (f2 a0 a1 = f2' a0 a1)) ∧
     (∀a0 a1. (M' = Pmult a0 a1) ⇒ (f3 a0 a1 = f3' a0 a1)) ∧
     (∀a. (M' = Popp a) ⇒ (f4 a = f4' a)) ⇒
     (polynom_CASE M f f1 f2 f3 f4 = polynom_CASE M' f' f1' f2' f3' f4')

polynom_nchotomy

|- ∀pp.
     (∃i. pp = Pvar i) ∨ (∃a. pp = Pconst a) ∨ (∃p p0. pp = Pplus p p0) ∨
     (∃p p0. pp = Pmult p p0) ∨ ∃p. pp = Popp p

polynom_Axiom

|- ∀f0 f1 f2 f3 f4.
     ∃fn.
       (∀a. fn (Pvar a) = f0 a) ∧ (∀a. fn (Pconst a) = f1 a) ∧
       (∀a0 a1. fn (Pplus a0 a1) = f2 a0 a1 (fn a0) (fn a1)) ∧
       (∀a0 a1. fn (Pmult a0 a1) = f3 a0 a1 (fn a0) (fn a1)) ∧
       ∀a. fn (Popp a) = f4 a (fn a)

polynom_induction

|- ∀P.
     (∀i. P (Pvar i)) ∧ (∀a. P (Pconst a)) ∧
     (∀p p0. P p ∧ P p0 ⇒ P (Pplus p p0)) ∧
     (∀p p0. P p ∧ P p0 ⇒ P (Pmult p p0)) ∧ (∀p. P p ⇒ P (Popp p)) ⇒
     ∀p. P p

polynom_normalize_ok

|- ∀r.
     is_ring r ⇒
     ∀vm p. r_interp_cs r vm (polynom_normalize r p) = interp_p r vm p

polynom_simplify_ok

|- ∀r.
     is_ring r ⇒
     ∀vm p. r_interp_cs r vm (polynom_simplify r p) = interp_p r vm p