Theory "canonical"

Parents semi_ring quote

Signature

Type	Arity
canonical_sum	1
spolynom	1
Constant	Type
Cons_monom	:α -> index list -> α canonical_sum -> α canonical_sum
Cons_varlist	:index list -> α canonical_sum -> α canonical_sum
Nil_monom	:α canonical_sum
SPconst	:α -> α spolynom
SPmult	:α spolynom -> α spolynom -> α spolynom
SPplus	:α spolynom -> α spolynom -> α spolynom
SPvar	:index -> α spolynom
canonical_sum_CASE	:α canonical_sum -> β -> (α -> index list -> α canonical_sum -> β) -> (index list -> α canonical_sum -> β) -> β
canonical_sum_merge	:α semi_ring -> α canonical_sum -> α canonical_sum -> α canonical_sum
canonical_sum_merge_tupled	:α semi_ring # α canonical_sum # α canonical_sum -> α canonical_sum
canonical_sum_prod	:α semi_ring -> α canonical_sum -> α canonical_sum -> α canonical_sum
canonical_sum_scalar	:α semi_ring -> α -> α canonical_sum -> α canonical_sum
canonical_sum_scalar2	:α semi_ring -> index list -> α canonical_sum -> α canonical_sum
canonical_sum_scalar3	:α semi_ring -> α -> index list -> α canonical_sum -> α canonical_sum
canonical_sum_simplify	:α semi_ring -> α canonical_sum -> α canonical_sum
canonical_sum_size	:(α -> num) -> α canonical_sum -> num
ics_aux	:α semi_ring -> α varmap -> α -> α canonical_sum -> α
interp_cs	:α semi_ring -> α varmap -> α canonical_sum -> α
interp_m	:α semi_ring -> α varmap -> α -> index list -> α
interp_sp	:α semi_ring -> α varmap -> α spolynom -> α
interp_vl	:α semi_ring -> α varmap -> index list -> α
ivl_aux	:α semi_ring -> α varmap -> index -> index list -> α
monom_insert	:α semi_ring -> α -> index list -> α canonical_sum -> α canonical_sum
monom_insert_tupled	:α semi_ring # α # index list # α canonical_sum -> α canonical_sum
spolynom_CASE	:α spolynom -> (index -> β) -> (α -> β) -> (α spolynom -> α spolynom -> β) -> (α spolynom -> α spolynom -> β) -> β
spolynom_normalize	:α semi_ring -> α spolynom -> α canonical_sum
spolynom_simplify	:α semi_ring -> α spolynom -> α canonical_sum
spolynom_size	:(α -> num) -> α spolynom -> num
varlist_insert	:α semi_ring -> index list -> α canonical_sum -> α canonical_sum
varlist_insert_tupled	:α semi_ring # index list # α canonical_sum -> α canonical_sum

Definitions

canonical_sum_TY_DEF

|- ∃rep.
     TYPE_DEFINITION
       (λa0'.
          ∀'canonical_sum' .
            (∀a0'.
               (a0' = ind_type$CONSTR 0 (ARB,ARB) (λn. ind_type$BOTTOM)) ∨
               (∃a0 a1 a2.
                  (a0' =
                   (λa0 a1 a2.
                      ind_type$CONSTR (SUC 0) (a0,a1)
                        (ind_type$FCONS a2 (λn. ind_type$BOTTOM))) a0 a1 a2) ∧
                  'canonical_sum' a2) ∨
               (∃a0 a1.
                  (a0' =
                   (λa0 a1.
                      ind_type$CONSTR (SUC (SUC 0)) (ARB,a0)
                        (ind_type$FCONS a1 (λn. ind_type$BOTTOM))) a0 a1) ∧
                  'canonical_sum' a1) ⇒
               'canonical_sum' a0') ⇒
            'canonical_sum' a0') rep

canonical_sum_case_def

|- (∀v f f1. canonical_sum_CASE Nil_monom v f f1 = v) ∧
   (∀a0 a1 a2 v f f1.
      canonical_sum_CASE (Cons_monom a0 a1 a2) v f f1 = f a0 a1 a2) ∧
   ∀a0 a1 v f f1. canonical_sum_CASE (Cons_varlist a0 a1) v f f1 = f1 a0 a1

canonical_sum_size_def

|- (∀f. canonical_sum_size f Nil_monom = 0) ∧
   (∀f a0 a1 a2.
      canonical_sum_size f (Cons_monom a0 a1 a2) =
      1 + (f a0 + (list_size index_size a1 + canonical_sum_size f a2))) ∧
   ∀f a0 a1.
     canonical_sum_size f (Cons_varlist a0 a1) =
     1 + (list_size index_size a0 + canonical_sum_size f a1)

canonical_sum_merge_tupled_primitive_def

|- canonical_sum_merge_tupled =
   WFREC
     (@R.
        WF R ∧
        (∀l2 c2 t2 t1 l1 c1 sr.
           R (sr,Cons_monom c1 l1 t1,t2)
             (sr,Cons_monom c1 l1 t1,Cons_monom c2 l2 t2)) ∧
        (∀l1 c1 t2 l2 c2 t1 sr.
           R (sr,t1,Cons_monom c2 l2 t2)
             (sr,Cons_monom c1 l1 t1,Cons_monom c2 l2 t2)) ∧
        (∀l2 c2 l1 c1 t2 t1 sr.
           R (sr,t1,t2) (sr,Cons_monom c1 l1 t1,Cons_monom c2 l2 t2)) ∧
        (∀l2 t2 t1 l1 c1 sr.
           R (sr,Cons_monom c1 l1 t1,t2)
             (sr,Cons_monom c1 l1 t1,Cons_varlist l2 t2)) ∧
        (∀l1 c1 t2 l2 t1 sr.
           R (sr,t1,Cons_varlist l2 t2)
             (sr,Cons_monom c1 l1 t1,Cons_varlist l2 t2)) ∧
        (∀l2 l1 c1 t2 t1 sr.
           R (sr,t1,t2) (sr,Cons_monom c1 l1 t1,Cons_varlist l2 t2)) ∧
        (∀l2 c2 t2 t1 l1 sr.
           R (sr,Cons_varlist l1 t1,t2)
             (sr,Cons_varlist l1 t1,Cons_monom c2 l2 t2)) ∧
        (∀l1 t2 l2 c2 t1 sr.
           R (sr,t1,Cons_monom c2 l2 t2)
             (sr,Cons_varlist l1 t1,Cons_monom c2 l2 t2)) ∧
        (∀l2 c2 l1 t2 t1 sr.
           R (sr,t1,t2) (sr,Cons_varlist l1 t1,Cons_monom c2 l2 t2)) ∧
        (∀l2 l1 t2 t1 sr.
           R (sr,t1,t2) (sr,Cons_varlist l1 t1,Cons_varlist l2 t2)) ∧
        (∀l1 t2 l2 t1 sr.
           R (sr,t1,Cons_varlist l2 t2)
             (sr,Cons_varlist l1 t1,Cons_varlist l2 t2)) ∧
        ∀l2 t2 t1 l1 sr.
          R (sr,Cons_varlist l1 t1,t2)
            (sr,Cons_varlist l1 t1,Cons_varlist l2 t2))
     (λcanonical_sum_merge_tupled a.
        case a of
          (sr,s1,Nil_monom) => I s1
        | (sr,Nil_monom,Cons_monom c2 l2 t2) => I (Cons_monom c2 l2 t2)
        | (sr,Cons_monom c1 l1 t1,Cons_monom c2 l2 t2) =>
            I
              (compare (list_compare index_compare l1 l2)
                 (Cons_monom c1 l1
                    (canonical_sum_merge_tupled (sr,t1,Cons_monom c2 l2 t2)))
                 (Cons_monom (SRP sr c1 c2) l1
                    (canonical_sum_merge_tupled (sr,t1,t2)))
                 (Cons_monom c2 l2
                    (canonical_sum_merge_tupled (sr,Cons_monom c1 l1 t1,t2))))
        | (sr,Cons_varlist l1'' t1'',Cons_monom c2 l2 t2) =>
            I
              (compare (list_compare index_compare l1'' l2)
                 (Cons_varlist l1''
                    (canonical_sum_merge_tupled
                       (sr,t1'',Cons_monom c2 l2 t2)))
                 (Cons_monom (SRP sr (SR1 sr) c2) l1''
                    (canonical_sum_merge_tupled (sr,t1'',t2)))
                 (Cons_monom c2 l2
                    (canonical_sum_merge_tupled
                       (sr,Cons_varlist l1'' t1'',t2))))
        | (sr,Nil_monom,Cons_varlist l2' t2') => I (Cons_varlist l2' t2')
        | (sr,Cons_monom c1' l1' t1',Cons_varlist l2' t2') =>
            I
              (compare (list_compare index_compare l1' l2')
                 (Cons_monom c1' l1'
                    (canonical_sum_merge_tupled
                       (sr,t1',Cons_varlist l2' t2')))
                 (Cons_monom (SRP sr c1' (SR1 sr)) l1'
                    (canonical_sum_merge_tupled (sr,t1',t2')))
                 (Cons_varlist l2'
                    (canonical_sum_merge_tupled
                       (sr,Cons_monom c1' l1' t1',t2'))))
        | (sr,Cons_varlist l1''' t1''',Cons_varlist l2' t2') =>
            I
              (compare (list_compare index_compare l1''' l2')
                 (Cons_varlist l1'''
                    (canonical_sum_merge_tupled
                       (sr,t1''',Cons_varlist l2' t2')))
                 (Cons_monom (SRP sr (SR1 sr) (SR1 sr)) l1'''
                    (canonical_sum_merge_tupled (sr,t1''',t2')))
                 (Cons_varlist l2'
                    (canonical_sum_merge_tupled
                       (sr,Cons_varlist l1''' t1''',t2')))))

canonical_sum_merge_curried_def

|- ∀x x1 x2.
     canonical_sum_merge x x1 x2 = canonical_sum_merge_tupled (x,x1,x2)

monom_insert_tupled_primitive_def

|- monom_insert_tupled =
   WFREC
     (@R.
        WF R ∧
        (∀l2 c2 t2 l1 c1 sr. R (sr,c1,l1,t2) (sr,c1,l1,Cons_monom c2 l2 t2)) ∧
        ∀l2 t2 l1 c1 sr. R (sr,c1,l1,t2) (sr,c1,l1,Cons_varlist l2 t2))
     (λmonom_insert_tupled a.
        case a of
          (sr,c1,l1,Nil_monom) => I (Cons_monom c1 l1 Nil_monom)
        | (sr,c1,l1,Cons_monom c2 l2 t2) =>
            I
              (compare (list_compare index_compare l1 l2)
                 (Cons_monom c1 l1 (Cons_monom c2 l2 t2))
                 (Cons_monom (SRP sr c1 c2) l1 t2)
                 (Cons_monom c2 l2 (monom_insert_tupled (sr,c1,l1,t2))))
        | (sr,c1,l1,Cons_varlist l2' t2') =>
            I
              (compare (list_compare index_compare l1 l2')
                 (Cons_monom c1 l1 (Cons_varlist l2' t2'))
                 (Cons_monom (SRP sr c1 (SR1 sr)) l1 t2')
                 (Cons_varlist l2' (monom_insert_tupled (sr,c1,l1,t2')))))

monom_insert_curried_def

|- ∀x x1 x2 x3. monom_insert x x1 x2 x3 = monom_insert_tupled (x,x1,x2,x3)

varlist_insert_tupled_primitive_def

|- varlist_insert_tupled =
   WFREC
     (@R.
        WF R ∧ (∀l2 c2 t2 l1 sr. R (sr,l1,t2) (sr,l1,Cons_monom c2 l2 t2)) ∧
        ∀l2 t2 l1 sr. R (sr,l1,t2) (sr,l1,Cons_varlist l2 t2))
     (λvarlist_insert_tupled a.
        case a of
          (sr,l1,Nil_monom) => I (Cons_varlist l1 Nil_monom)
        | (sr,l1,Cons_monom c2 l2 t2) =>
            I
              (compare (list_compare index_compare l1 l2)
                 (Cons_varlist l1 (Cons_monom c2 l2 t2))
                 (Cons_monom (SRP sr (SR1 sr) c2) l1 t2)
                 (Cons_monom c2 l2 (varlist_insert_tupled (sr,l1,t2))))
        | (sr,l1,Cons_varlist l2' t2') =>
            I
              (compare (list_compare index_compare l1 l2')
                 (Cons_varlist l1 (Cons_varlist l2' t2'))
                 (Cons_monom (SRP sr (SR1 sr) (SR1 sr)) l1 t2')
                 (Cons_varlist l2' (varlist_insert_tupled (sr,l1,t2')))))

varlist_insert_curried_def

|- ∀x x1 x2. varlist_insert x x1 x2 = varlist_insert_tupled (x,x1,x2)

canonical_sum_scalar_def

|- (∀sr c0 c l t.
      canonical_sum_scalar sr c0 (Cons_monom c l t) =
      Cons_monom (SRM sr c0 c) l (canonical_sum_scalar sr c0 t)) ∧
   (∀sr c0 l t.
      canonical_sum_scalar sr c0 (Cons_varlist l t) =
      Cons_monom c0 l (canonical_sum_scalar sr c0 t)) ∧
   ∀sr c0. canonical_sum_scalar sr c0 Nil_monom = Nil_monom

canonical_sum_scalar2_def

|- (∀sr l0 c l t.
      canonical_sum_scalar2 sr l0 (Cons_monom c l t) =
      monom_insert sr c (list_merge index_lt l0 l)
        (canonical_sum_scalar2 sr l0 t)) ∧
   (∀sr l0 l t.
      canonical_sum_scalar2 sr l0 (Cons_varlist l t) =
      varlist_insert sr (list_merge index_lt l0 l)
        (canonical_sum_scalar2 sr l0 t)) ∧
   ∀sr l0. canonical_sum_scalar2 sr l0 Nil_monom = Nil_monom

canonical_sum_scalar3_def

|- (∀sr c0 l0 c l t.
      canonical_sum_scalar3 sr c0 l0 (Cons_monom c l t) =
      monom_insert sr (SRM sr c0 c) (list_merge index_lt l0 l)
        (canonical_sum_scalar3 sr c0 l0 t)) ∧
   (∀sr c0 l0 l t.
      canonical_sum_scalar3 sr c0 l0 (Cons_varlist l t) =
      monom_insert sr c0 (list_merge index_lt l0 l)
        (canonical_sum_scalar3 sr c0 l0 t)) ∧
   ∀sr c0 l0. canonical_sum_scalar3 sr c0 l0 Nil_monom = Nil_monom

canonical_sum_prod_def

|- (∀sr c1 l1 t1 s2.
      canonical_sum_prod sr (Cons_monom c1 l1 t1) s2 =
      canonical_sum_merge sr (canonical_sum_scalar3 sr c1 l1 s2)
        (canonical_sum_prod sr t1 s2)) ∧
   (∀sr l1 t1 s2.
      canonical_sum_prod sr (Cons_varlist l1 t1) s2 =
      canonical_sum_merge sr (canonical_sum_scalar2 sr l1 s2)
        (canonical_sum_prod sr t1 s2)) ∧
   ∀sr s2. canonical_sum_prod sr Nil_monom s2 = Nil_monom

canonical_sum_simplify_def

|- (∀sr c l t.
      canonical_sum_simplify sr (Cons_monom c l t) =
      if c = SR0 sr then canonical_sum_simplify sr t
      else if c = SR1 sr then Cons_varlist l (canonical_sum_simplify sr t)
      else Cons_monom c l (canonical_sum_simplify sr t)) ∧
   (∀sr l t.
      canonical_sum_simplify sr (Cons_varlist l t) =
      Cons_varlist l (canonical_sum_simplify sr t)) ∧
   ∀sr. canonical_sum_simplify sr Nil_monom = Nil_monom

ivl_aux_def

|- (∀sr vm x. ivl_aux sr vm x [] = varmap_find x vm) ∧
   ∀sr vm x x' t'.
     ivl_aux sr vm x (x'::t') =
     SRM sr (varmap_find x vm) (ivl_aux sr vm x' t')

interp_vl_def

|- (∀sr vm. interp_vl sr vm [] = SR1 sr) ∧
   ∀sr vm x t. interp_vl sr vm (x::t) = ivl_aux sr vm x t

interp_m_def

|- (∀sr vm c. interp_m sr vm c [] = c) ∧
   ∀sr vm c x t. interp_m sr vm c (x::t) = SRM sr c (ivl_aux sr vm x t)

ics_aux_def

|- (∀sr vm a. ics_aux sr vm a Nil_monom = a) ∧
   (∀sr vm a l t.
      ics_aux sr vm a (Cons_varlist l t) =
      SRP sr a (ics_aux sr vm (interp_vl sr vm l) t)) ∧
   ∀sr vm a c l t.
     ics_aux sr vm a (Cons_monom c l t) =
     SRP sr a (ics_aux sr vm (interp_m sr vm c l) t)

interp_cs_def

|- (∀sr vm. interp_cs sr vm Nil_monom = SR0 sr) ∧
   (∀sr vm l t.
      interp_cs sr vm (Cons_varlist l t) =
      ics_aux sr vm (interp_vl sr vm l) t) ∧
   ∀sr vm c l t.
     interp_cs sr vm (Cons_monom c l t) = ics_aux sr vm (interp_m sr vm c l) t

spolynom_TY_DEF

|- ∃rep.
     TYPE_DEFINITION
       (λa0'.
          ∀'spolynom' .
            (∀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) ∧ 'spolynom' a0 ∧ 'spolynom' 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) ∧ 'spolynom' a0 ∧ 'spolynom' a1) ⇒
               'spolynom' a0') ⇒
            'spolynom' a0') rep

spolynom_case_def

|- (∀a f f1 f2 f3. spolynom_CASE (SPvar a) f f1 f2 f3 = f a) ∧
   (∀a f f1 f2 f3. spolynom_CASE (SPconst a) f f1 f2 f3 = f1 a) ∧
   (∀a0 a1 f f1 f2 f3. spolynom_CASE (SPplus a0 a1) f f1 f2 f3 = f2 a0 a1) ∧
   ∀a0 a1 f f1 f2 f3. spolynom_CASE (SPmult a0 a1) f f1 f2 f3 = f3 a0 a1

spolynom_size_def

|- (∀f a. spolynom_size f (SPvar a) = 1 + index_size a) ∧
   (∀f a. spolynom_size f (SPconst a) = 1 + f a) ∧
   (∀f a0 a1.
      spolynom_size f (SPplus a0 a1) =
      1 + (spolynom_size f a0 + spolynom_size f a1)) ∧
   ∀f a0 a1.
     spolynom_size f (SPmult a0 a1) =
     1 + (spolynom_size f a0 + spolynom_size f a1)

spolynom_normalize_def

|- (∀sr i. spolynom_normalize sr (SPvar i) = Cons_varlist [i] Nil_monom) ∧
   (∀sr c. spolynom_normalize sr (SPconst c) = Cons_monom c [] Nil_monom) ∧
   (∀sr l r.
      spolynom_normalize sr (SPplus l r) =
      canonical_sum_merge sr (spolynom_normalize sr l)
        (spolynom_normalize sr r)) ∧
   ∀sr l r.
     spolynom_normalize sr (SPmult l r) =
     canonical_sum_prod sr (spolynom_normalize sr l) (spolynom_normalize sr r)

spolynom_simplify_def

|- ∀sr x.
     spolynom_simplify sr x =
     canonical_sum_simplify sr (spolynom_normalize sr x)

interp_sp_def

|- (∀sr vm c. interp_sp sr vm (SPconst c) = c) ∧
   (∀sr vm i. interp_sp sr vm (SPvar i) = varmap_find i vm) ∧
   (∀sr vm p1 p2.
      interp_sp sr vm (SPplus p1 p2) =
      SRP sr (interp_sp sr vm p1) (interp_sp sr vm p2)) ∧
   ∀sr vm p1 p2.
     interp_sp sr vm (SPmult p1 p2) =
     SRM sr (interp_sp sr vm p1) (interp_sp sr vm p2)

Theorems

datatype_canonical_sum

|- DATATYPE (canonical_sum Nil_monom Cons_monom Cons_varlist)

canonical_sum_11

|- (∀a0 a1 a2 a0' a1' a2'.
      (Cons_monom a0 a1 a2 = Cons_monom a0' a1' a2') ⇔
      (a0 = a0') ∧ (a1 = a1') ∧ (a2 = a2')) ∧
   ∀a0 a1 a0' a1'.
     (Cons_varlist a0 a1 = Cons_varlist a0' a1') ⇔ (a0 = a0') ∧ (a1 = a1')

canonical_sum_distinct

|- (∀a2 a1 a0. Nil_monom ≠ Cons_monom a0 a1 a2) ∧
   (∀a1 a0. Nil_monom ≠ Cons_varlist a0 a1) ∧
   ∀a2 a1' a1 a0' a0. Cons_monom a0 a1 a2 ≠ Cons_varlist a0' a1'

canonical_sum_case_cong

|- ∀M M' v f f1.
     (M = M') ∧ ((M' = Nil_monom) ⇒ (v = v')) ∧
     (∀a0 a1 a2. (M' = Cons_monom a0 a1 a2) ⇒ (f a0 a1 a2 = f' a0 a1 a2)) ∧
     (∀a0 a1. (M' = Cons_varlist a0 a1) ⇒ (f1 a0 a1 = f1' a0 a1)) ⇒
     (canonical_sum_CASE M v f f1 = canonical_sum_CASE M' v' f' f1')

canonical_sum_nchotomy

|- ∀cc.
     (cc = Nil_monom) ∨ (∃a l c. cc = Cons_monom a l c) ∨
     ∃l c. cc = Cons_varlist l c

canonical_sum_Axiom

|- ∀f0 f1 f2.
     ∃fn.
       (fn Nil_monom = f0) ∧
       (∀a0 a1 a2. fn (Cons_monom a0 a1 a2) = f1 a0 a1 a2 (fn a2)) ∧
       ∀a0 a1. fn (Cons_varlist a0 a1) = f2 a0 a1 (fn a1)

canonical_sum_induction

|- ∀P.
     P Nil_monom ∧ (∀c. P c ⇒ ∀l a. P (Cons_monom a l c)) ∧
     (∀c. P c ⇒ ∀l. P (Cons_varlist l c)) ⇒
     ∀c. P c

canonical_sum_merge_ind

|- ∀P.
     (∀sr c1 l1 t1 c2 l2 t2.
        P sr t1 t2 ∧ P sr t1 (Cons_monom c2 l2 t2) ∧
        P sr (Cons_monom c1 l1 t1) t2 ⇒
        P sr (Cons_monom c1 l1 t1) (Cons_monom c2 l2 t2)) ∧
     (∀sr c1 l1 t1 l2 t2.
        P sr t1 t2 ∧ P sr t1 (Cons_varlist l2 t2) ∧
        P sr (Cons_monom c1 l1 t1) t2 ⇒
        P sr (Cons_monom c1 l1 t1) (Cons_varlist l2 t2)) ∧
     (∀sr l1 t1 c2 l2 t2.
        P sr t1 t2 ∧ P sr t1 (Cons_monom c2 l2 t2) ∧
        P sr (Cons_varlist l1 t1) t2 ⇒
        P sr (Cons_varlist l1 t1) (Cons_monom c2 l2 t2)) ∧
     (∀sr l1 t1 l2 t2.
        P sr t1 t2 ∧ P sr t1 (Cons_varlist l2 t2) ∧
        P sr (Cons_varlist l1 t1) t2 ⇒
        P sr (Cons_varlist l1 t1) (Cons_varlist l2 t2)) ∧
     (∀sr s1. P sr s1 Nil_monom) ∧
     (∀sr v4 v5 v6. P sr Nil_monom (Cons_monom v4 v5 v6)) ∧
     (∀sr v7 v8. P sr Nil_monom (Cons_varlist v7 v8)) ⇒
     ∀v v1 v2. P v v1 v2

canonical_sum_merge_def

|- (∀t2 t1 sr l2 l1 c2 c1.
      canonical_sum_merge sr (Cons_monom c1 l1 t1) (Cons_monom c2 l2 t2) =
      compare (list_compare index_compare l1 l2)
        (Cons_monom c1 l1 (canonical_sum_merge sr t1 (Cons_monom c2 l2 t2)))
        (Cons_monom (SRP sr c1 c2) l1 (canonical_sum_merge sr t1 t2))
        (Cons_monom c2 l2
           (canonical_sum_merge sr (Cons_monom c1 l1 t1) t2))) ∧
   (∀t2 t1 sr l2 l1 c1.
      canonical_sum_merge sr (Cons_monom c1 l1 t1) (Cons_varlist l2 t2) =
      compare (list_compare index_compare l1 l2)
        (Cons_monom c1 l1 (canonical_sum_merge sr t1 (Cons_varlist l2 t2)))
        (Cons_monom (SRP sr c1 (SR1 sr)) l1 (canonical_sum_merge sr t1 t2))
        (Cons_varlist l2 (canonical_sum_merge sr (Cons_monom c1 l1 t1) t2))) ∧
   (∀t2 t1 sr l2 l1 c2.
      canonical_sum_merge sr (Cons_varlist l1 t1) (Cons_monom c2 l2 t2) =
      compare (list_compare index_compare l1 l2)
        (Cons_varlist l1 (canonical_sum_merge sr t1 (Cons_monom c2 l2 t2)))
        (Cons_monom (SRP sr (SR1 sr) c2) l1 (canonical_sum_merge sr t1 t2))
        (Cons_monom c2 l2 (canonical_sum_merge sr (Cons_varlist l1 t1) t2))) ∧
   (∀t2 t1 sr l2 l1.
      canonical_sum_merge sr (Cons_varlist l1 t1) (Cons_varlist l2 t2) =
      compare (list_compare index_compare l1 l2)
        (Cons_varlist l1 (canonical_sum_merge sr t1 (Cons_varlist l2 t2)))
        (Cons_monom (SRP sr (SR1 sr) (SR1 sr)) l1
           (canonical_sum_merge sr t1 t2))
        (Cons_varlist l2 (canonical_sum_merge sr (Cons_varlist l1 t1) t2))) ∧
   (∀sr s1. canonical_sum_merge sr s1 Nil_monom = s1) ∧
   (∀v6 v5 v4 sr.
      canonical_sum_merge sr Nil_monom (Cons_monom v4 v5 v6) =
      Cons_monom v4 v5 v6) ∧
   ∀v8 v7 sr.
     canonical_sum_merge sr Nil_monom (Cons_varlist v7 v8) =
     Cons_varlist v7 v8

monom_insert_ind

|- ∀P.
     (∀sr c1 l1 c2 l2 t2. P sr c1 l1 t2 ⇒ P sr c1 l1 (Cons_monom c2 l2 t2)) ∧
     (∀sr c1 l1 l2 t2. P sr c1 l1 t2 ⇒ P sr c1 l1 (Cons_varlist l2 t2)) ∧
     (∀sr c1 l1. P sr c1 l1 Nil_monom) ⇒
     ∀v v1 v2 v3. P v v1 v2 v3

monom_insert_def

|- (∀t2 sr l2 l1 c2 c1.
      monom_insert sr 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 (SRP sr c1 c2) l1 t2)
        (Cons_monom c2 l2 (monom_insert sr c1 l1 t2))) ∧
   (∀t2 sr l2 l1 c1.
      monom_insert sr c1 l1 (Cons_varlist l2 t2) =
      compare (list_compare index_compare l1 l2)
        (Cons_monom c1 l1 (Cons_varlist l2 t2))
        (Cons_monom (SRP sr c1 (SR1 sr)) l1 t2)
        (Cons_varlist l2 (monom_insert sr c1 l1 t2))) ∧
   ∀sr l1 c1. monom_insert sr c1 l1 Nil_monom = Cons_monom c1 l1 Nil_monom

varlist_insert_ind

|- ∀P.
     (∀sr l1 c2 l2 t2. P sr l1 t2 ⇒ P sr l1 (Cons_monom c2 l2 t2)) ∧
     (∀sr l1 l2 t2. P sr l1 t2 ⇒ P sr l1 (Cons_varlist l2 t2)) ∧
     (∀sr l1. P sr l1 Nil_monom) ⇒
     ∀v v1 v2. P v v1 v2

varlist_insert_def

|- (∀t2 sr l2 l1 c2.
      varlist_insert sr l1 (Cons_monom c2 l2 t2) =
      compare (list_compare index_compare l1 l2)
        (Cons_varlist l1 (Cons_monom c2 l2 t2))
        (Cons_monom (SRP sr (SR1 sr) c2) l1 t2)
        (Cons_monom c2 l2 (varlist_insert sr l1 t2))) ∧
   (∀t2 sr l2 l1.
      varlist_insert sr l1 (Cons_varlist l2 t2) =
      compare (list_compare index_compare l1 l2)
        (Cons_varlist l1 (Cons_varlist l2 t2))
        (Cons_monom (SRP sr (SR1 sr) (SR1 sr)) l1 t2)
        (Cons_varlist l2 (varlist_insert sr l1 t2))) ∧
   ∀sr l1. varlist_insert sr l1 Nil_monom = Cons_varlist l1 Nil_monom

ivl_aux_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm v i.
       ivl_aux sr vm i v = SRM sr (varmap_find i vm) (interp_vl sr vm v)

varlist_merge_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm x y.
       interp_vl sr vm (list_merge index_lt x y) =
       SRM sr (interp_vl sr vm x) (interp_vl sr vm y)

ics_aux_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm x s. ics_aux sr vm x s = SRP sr x (interp_cs sr vm s)

interp_m_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm x l. interp_m sr vm x l = SRM sr x (interp_vl sr vm l)

canonical_sum_merge_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm x y.
       interp_cs sr vm (canonical_sum_merge sr x y) =
       SRP sr (interp_cs sr vm x) (interp_cs sr vm y)

monom_insert_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm a l s.
       interp_cs sr vm (monom_insert sr a l s) =
       SRP sr (SRM sr a (interp_vl sr vm l)) (interp_cs sr vm s)

varlist_insert_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm l s.
       interp_cs sr vm (varlist_insert sr l s) =
       SRP sr (interp_vl sr vm l) (interp_cs sr vm s)

canonical_sum_scalar_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm a s.
       interp_cs sr vm (canonical_sum_scalar sr a s) =
       SRM sr a (interp_cs sr vm s)

canonical_sum_scalar2_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm l s.
       interp_cs sr vm (canonical_sum_scalar2 sr l s) =
       SRM sr (interp_vl sr vm l) (interp_cs sr vm s)

canonical_sum_scalar3_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm c l s.
       interp_cs sr vm (canonical_sum_scalar3 sr c l s) =
       SRM sr (SRM sr c (interp_vl sr vm l)) (interp_cs sr vm s)

canonical_sum_prod_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm x y.
       interp_cs sr vm (canonical_sum_prod sr x y) =
       SRM sr (interp_cs sr vm x) (interp_cs sr vm y)

canonical_sum_simplify_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm s. interp_cs sr vm (canonical_sum_simplify sr s) = interp_cs sr vm s

datatype_spolynom

|- DATATYPE (spolynom SPvar SPconst SPplus SPmult)

spolynom_11

|- (∀a a'. (SPvar a = SPvar a') ⇔ (a = a')) ∧
   (∀a a'. (SPconst a = SPconst a') ⇔ (a = a')) ∧
   (∀a0 a1 a0' a1'.
      (SPplus a0 a1 = SPplus a0' a1') ⇔ (a0 = a0') ∧ (a1 = a1')) ∧
   ∀a0 a1 a0' a1'. (SPmult a0 a1 = SPmult a0' a1') ⇔ (a0 = a0') ∧ (a1 = a1')

spolynom_distinct

|- (∀a' a. SPvar a ≠ SPconst a') ∧ (∀a1 a0 a. SPvar a ≠ SPplus a0 a1) ∧
   (∀a1 a0 a. SPvar a ≠ SPmult a0 a1) ∧ (∀a1 a0 a. SPconst a ≠ SPplus a0 a1) ∧
   (∀a1 a0 a. SPconst a ≠ SPmult a0 a1) ∧
   ∀a1' a1 a0' a0. SPplus a0 a1 ≠ SPmult a0' a1'

spolynom_case_cong

|- ∀M M' f f1 f2 f3.
     (M = M') ∧ (∀a. (M' = SPvar a) ⇒ (f a = f' a)) ∧
     (∀a. (M' = SPconst a) ⇒ (f1 a = f1' a)) ∧
     (∀a0 a1. (M' = SPplus a0 a1) ⇒ (f2 a0 a1 = f2' a0 a1)) ∧
     (∀a0 a1. (M' = SPmult a0 a1) ⇒ (f3 a0 a1 = f3' a0 a1)) ⇒
     (spolynom_CASE M f f1 f2 f3 = spolynom_CASE M' f' f1' f2' f3')

spolynom_nchotomy

|- ∀ss.
     (∃i. ss = SPvar i) ∨ (∃a. ss = SPconst a) ∨ (∃s s0. ss = SPplus s s0) ∨
     ∃s s0. ss = SPmult s s0

spolynom_Axiom

|- ∀f0 f1 f2 f3.
     ∃fn.
       (∀a. fn (SPvar a) = f0 a) ∧ (∀a. fn (SPconst a) = f1 a) ∧
       (∀a0 a1. fn (SPplus a0 a1) = f2 a0 a1 (fn a0) (fn a1)) ∧
       ∀a0 a1. fn (SPmult a0 a1) = f3 a0 a1 (fn a0) (fn a1)

spolynom_induction

|- ∀P.
     (∀i. P (SPvar i)) ∧ (∀a. P (SPconst a)) ∧
     (∀s s0. P s ∧ P s0 ⇒ P (SPplus s s0)) ∧
     (∀s s0. P s ∧ P s0 ⇒ P (SPmult s s0)) ⇒
     ∀s. P s

spolynomial_normalize_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm p. interp_cs sr vm (spolynom_normalize sr p) = interp_sp sr vm p

spolynomial_simplify_ok

|- ∀sr.
     is_semi_ring sr ⇒
     ∀vm p. interp_cs sr vm (spolynom_simplify sr p) = interp_sp sr vm p