Theory "semi_ring"

Parents list

Signature

Type	Arity
semi_ring	1
Constant	Type
is_semi_ring	:α semi_ring -> bool
semi_ring	:α -> α -> (α -> α -> α) -> (α -> α -> α) -> α semi_ring
semi_ring_CASE	:α semi_ring -> (α -> α -> (α -> α -> α) -> (α -> α -> α) -> β) -> β
semi_ring_SR0	:α semi_ring -> α
semi_ring_SR0_fupd	:(α -> α) -> α semi_ring -> α semi_ring
semi_ring_SR1	:α semi_ring -> α
semi_ring_SR1_fupd	:(α -> α) -> α semi_ring -> α semi_ring
semi_ring_SRM	:α semi_ring -> α -> α -> α
semi_ring_SRM_fupd	:((α -> α -> α) -> α -> α -> α) -> α semi_ring -> α semi_ring
semi_ring_SRP	:α semi_ring -> α -> α -> α
semi_ring_SRP_fupd	:((α -> α -> α) -> α -> α -> α) -> α semi_ring -> α semi_ring
semi_ring_size	:(α -> num) -> α semi_ring -> num

Definitions

semi_ring_TY_DEF

|- ∃rep.
     TYPE_DEFINITION
       (λa0'.
          ∀'semi_ring' .
            (∀a0'.
               (∃a0 a1 a2 a3.
                  a0' =
                  (λa0 a1 a2 a3.
                     ind_type$CONSTR 0 (a0,a1,a2,a3) (λn. ind_type$BOTTOM)) a0
                    a1 a2 a3) ⇒
               'semi_ring' a0') ⇒
            'semi_ring' a0') rep

semi_ring_case_def

|- ∀a0 a1 a2 a3 f. semi_ring_CASE (semi_ring a0 a1 a2 a3) f = f a0 a1 a2 a3

semi_ring_size_def

|- ∀f a0 a1 a2 a3.
     semi_ring_size f (semi_ring a0 a1 a2 a3) = 1 + (f a0 + f a1)

semi_ring_SR0

|- ∀a a0 f f0. SR0 (semi_ring a a0 f f0) = a

semi_ring_SR1

|- ∀a a0 f f0. SR1 (semi_ring a a0 f f0) = a0

semi_ring_SRP

|- ∀a a0 f f0. SRP (semi_ring a a0 f f0) = f

semi_ring_SRM

|- ∀a a0 f f0. SRM (semi_ring a a0 f f0) = f0

semi_ring_SR0_fupd

|- ∀f1 a a0 f f0.
     semi_ring a a0 f f0 with SR0 updated_by f1 = semi_ring (f1 a) a0 f f0

semi_ring_SR1_fupd

|- ∀f1 a a0 f f0.
     semi_ring a a0 f f0 with SR1 updated_by f1 = semi_ring a (f1 a0) f f0

semi_ring_SRP_fupd

|- ∀f1 a a0 f f0.
     semi_ring a a0 f f0 with SRP updated_by f1 = semi_ring a a0 (f1 f) f0

semi_ring_SRM_fupd

|- ∀f1 a a0 f f0.
     semi_ring a a0 f f0 with SRM updated_by f1 = semi_ring a a0 f (f1 f0)

is_semi_ring_def

|- ∀r.
     is_semi_ring r ⇔
     (∀n m. SRP r n m = SRP r m n) ∧
     (∀n m p. SRP r n (SRP r m p) = SRP r (SRP r n m) p) ∧
     (∀n m. SRM r n m = SRM r m n) ∧
     (∀n m p. SRM r n (SRM r m p) = SRM r (SRM r n m) p) ∧
     (∀n. SRP r (SR0 r) n = n) ∧ (∀n. SRM r (SR1 r) n = n) ∧
     (∀n. SRM r (SR0 r) n = SR0 r) ∧
     ∀n m p. SRM r (SRP r n m) p = SRP r (SRM r n p) (SRM r m p)

Theorems

semi_ring_accessors

|- (∀a a0 f f0. SR0 (semi_ring a a0 f f0) = a) ∧
   (∀a a0 f f0. SR1 (semi_ring a a0 f f0) = a0) ∧
   (∀a a0 f f0. SRP (semi_ring a a0 f f0) = f) ∧
   ∀a a0 f f0. SRM (semi_ring a a0 f f0) = f0

semi_ring_fn_updates

|- (∀f1 a a0 f f0.
      semi_ring a a0 f f0 with SR0 updated_by f1 = semi_ring (f1 a) a0 f f0) ∧
   (∀f1 a a0 f f0.
      semi_ring a a0 f f0 with SR1 updated_by f1 = semi_ring a (f1 a0) f f0) ∧
   (∀f1 a a0 f f0.
      semi_ring a a0 f f0 with SRP updated_by f1 = semi_ring a a0 (f1 f) f0) ∧
   ∀f1 a a0 f f0.
     semi_ring a a0 f f0 with SRM updated_by f1 = semi_ring a a0 f (f1 f0)

semi_ring_accfupds

|- (∀s f. SR0 (s with SR1 updated_by f) = SR0 s) ∧
   (∀s f. SR0 (s with SRP updated_by f) = SR0 s) ∧
   (∀s f. SR0 (s with SRM updated_by f) = SR0 s) ∧
   (∀s f. SR1 (s with SR0 updated_by f) = SR1 s) ∧
   (∀s f. SR1 (s with SRP updated_by f) = SR1 s) ∧
   (∀s f. SR1 (s with SRM updated_by f) = SR1 s) ∧
   (∀s f. SRP (s with SR0 updated_by f) = SRP s) ∧
   (∀s f. SRP (s with SR1 updated_by f) = SRP s) ∧
   (∀s f. SRP (s with SRM updated_by f) = SRP s) ∧
   (∀s f. SRM (s with SR0 updated_by f) = SRM s) ∧
   (∀s f. SRM (s with SR1 updated_by f) = SRM s) ∧
   (∀s f. SRM (s with SRP updated_by f) = SRM s) ∧
   (∀s f. SR0 (s with SR0 updated_by f) = f (SR0 s)) ∧
   (∀s f. SR1 (s with SR1 updated_by f) = f (SR1 s)) ∧
   (∀s f. SRP (s with SRP updated_by f) = f (SRP s)) ∧
   ∀s f. SRM (s with SRM updated_by f) = f (SRM s)

semi_ring_fupdfupds

|- (∀s g f.
      s with <|SR0 updated_by f; SR0 updated_by g|> =
      s with SR0 updated_by f o g) ∧
   (∀s g f.
      s with <|SR1 updated_by f; SR1 updated_by g|> =
      s with SR1 updated_by f o g) ∧
   (∀s g f.
      s with <|SRP updated_by f; SRP updated_by g|> =
      s with SRP updated_by f o g) ∧
   ∀s g f.
     s with <|SRM updated_by f; SRM updated_by g|> =
     s with SRM updated_by f o g

semi_ring_fupdfupds_comp

|- ((∀g f.
        _ record fupdateSR0 f o  _ record fupdateSR0 g =
        _ record fupdateSR0 (f o g)) ∧
    ∀h g f.
       _ record fupdateSR0 f o  _ record fupdateSR0 g o h =
       _ record fupdateSR0 (f o g) o h) ∧
   ((∀g f.
        _ record fupdateSR1 f o  _ record fupdateSR1 g =
        _ record fupdateSR1 (f o g)) ∧
    ∀h g f.
       _ record fupdateSR1 f o  _ record fupdateSR1 g o h =
       _ record fupdateSR1 (f o g) o h) ∧
   ((∀g f.
        _ record fupdateSRP f o  _ record fupdateSRP g =
        _ record fupdateSRP (f o g)) ∧
    ∀h g f.
       _ record fupdateSRP f o  _ record fupdateSRP g o h =
       _ record fupdateSRP (f o g) o h) ∧
   (∀g f.
       _ record fupdateSRM f o  _ record fupdateSRM g =
       _ record fupdateSRM (f o g)) ∧
   ∀h g f.
      _ record fupdateSRM f o  _ record fupdateSRM g o h =
      _ record fupdateSRM (f o g) o h

semi_ring_fupdcanon

|- (∀s g f.
      s with <|SR1 updated_by f; SR0 updated_by g|> =
      s with <|SR0 updated_by g; SR1 updated_by f|>) ∧
   (∀s g f.
      s with <|SRP updated_by f; SR0 updated_by g|> =
      s with <|SR0 updated_by g; SRP updated_by f|>) ∧
   (∀s g f.
      s with <|SRP updated_by f; SR1 updated_by g|> =
      s with <|SR1 updated_by g; SRP updated_by f|>) ∧
   (∀s g f.
      s with <|SRM updated_by f; SR0 updated_by g|> =
      s with <|SR0 updated_by g; SRM updated_by f|>) ∧
   (∀s g f.
      s with <|SRM updated_by f; SR1 updated_by g|> =
      s with <|SR1 updated_by g; SRM updated_by f|>) ∧
   ∀s g f.
     s with <|SRM updated_by f; SRP updated_by g|> =
     s with <|SRP updated_by g; SRM updated_by f|>

semi_ring_fupdcanon_comp

|- ((∀g f.
        _ record fupdateSR1 f o  _ record fupdateSR0 g =
        _ record fupdateSR0 g o  _ record fupdateSR1 f) ∧
    ∀h g f.
       _ record fupdateSR1 f o  _ record fupdateSR0 g o h =
       _ record fupdateSR0 g o  _ record fupdateSR1 f o h) ∧
   ((∀g f.
        _ record fupdateSRP f o  _ record fupdateSR0 g =
        _ record fupdateSR0 g o  _ record fupdateSRP f) ∧
    ∀h g f.
       _ record fupdateSRP f o  _ record fupdateSR0 g o h =
       _ record fupdateSR0 g o  _ record fupdateSRP f o h) ∧
   ((∀g f.
        _ record fupdateSRP f o  _ record fupdateSR1 g =
        _ record fupdateSR1 g o  _ record fupdateSRP f) ∧
    ∀h g f.
       _ record fupdateSRP f o  _ record fupdateSR1 g o h =
       _ record fupdateSR1 g o  _ record fupdateSRP f o h) ∧
   ((∀g f.
        _ record fupdateSRM f o  _ record fupdateSR0 g =
        _ record fupdateSR0 g o  _ record fupdateSRM f) ∧
    ∀h g f.
       _ record fupdateSRM f o  _ record fupdateSR0 g o h =
       _ record fupdateSR0 g o  _ record fupdateSRM f o h) ∧
   ((∀g f.
        _ record fupdateSRM f o  _ record fupdateSR1 g =
        _ record fupdateSR1 g o  _ record fupdateSRM f) ∧
    ∀h g f.
       _ record fupdateSRM f o  _ record fupdateSR1 g o h =
       _ record fupdateSR1 g o  _ record fupdateSRM f o h) ∧
   (∀g f.
       _ record fupdateSRM f o  _ record fupdateSRP g =
       _ record fupdateSRP g o  _ record fupdateSRM f) ∧
   ∀h g f.
      _ record fupdateSRM f o  _ record fupdateSRP g o h =
      _ record fupdateSRP g o  _ record fupdateSRM f o h

semi_ring_component_equality

|- ∀s1 s2.
     (s1 = s2) ⇔
     (SR0 s1 = SR0 s2) ∧ (SR1 s1 = SR1 s2) ∧ (SRP s1 = SRP s2) ∧
     (SRM s1 = SRM s2)

semi_ring_updates_eq_literal

|- ∀s a0 a f0 f.
     s with <|SR0 := a0; SR1 := a; SRP := f0; SRM := f|> =
     <|SR0 := a0; SR1 := a; SRP := f0; SRM := f|>

semi_ring_literal_nchotomy

|- ∀s. ∃a0 a f0 f. s = <|SR0 := a0; SR1 := a; SRP := f0; SRM := f|>

FORALL_semi_ring

|- ∀P. (∀s. P s) ⇔ ∀a0 a f0 f. P <|SR0 := a0; SR1 := a; SRP := f0; SRM := f|>

EXISTS_semi_ring

|- ∀P. (∃s. P s) ⇔ ∃a0 a f0 f. P <|SR0 := a0; SR1 := a; SRP := f0; SRM := f|>

semi_ring_literal_11

|- ∀a01 a1 f01 f1 a02 a2 f02 f2.
     (<|SR0 := a01; SR1 := a1; SRP := f01; SRM := f1|> =
      <|SR0 := a02; SR1 := a2; SRP := f02; SRM := f2|>) ⇔
     (a01 = a02) ∧ (a1 = a2) ∧ (f01 = f02) ∧ (f1 = f2)

datatype_semi_ring

|- DATATYPE (record semi_ring SR0 SR1 SRP SRM)

semi_ring_11

|- ∀a0 a1 a2 a3 a0' a1' a2' a3'.
     (semi_ring a0 a1 a2 a3 = semi_ring a0' a1' a2' a3') ⇔
     (a0 = a0') ∧ (a1 = a1') ∧ (a2 = a2') ∧ (a3 = a3')

semi_ring_case_cong

|- ∀M M' f.
     (M = M') ∧
     (∀a0 a1 a2 a3.
        (M' = semi_ring a0 a1 a2 a3) ⇒ (f a0 a1 a2 a3 = f' a0 a1 a2 a3)) ⇒
     (semi_ring_CASE M f = semi_ring_CASE M' f')

semi_ring_nchotomy

|- ∀ss. ∃a a0 f f0. ss = semi_ring a a0 f f0

semi_ring_Axiom

|- ∀f. ∃fn. ∀a0 a1 a2 a3. fn (semi_ring a0 a1 a2 a3) = f a0 a1 a2 a3

semi_ring_induction

|- ∀P. (∀a a0 f f0. P (semi_ring a a0 f f0)) ⇒ ∀s. P s

plus_sym

|- ∀r. is_semi_ring r ⇒ ∀n m. SRP r n m = SRP r m n

plus_assoc

|- ∀r. is_semi_ring r ⇒ ∀n m p. SRP r n (SRP r m p) = SRP r (SRP r n m) p

mult_sym

|- ∀r. is_semi_ring r ⇒ ∀n m. SRM r n m = SRM r m n

mult_assoc

|- ∀r. is_semi_ring r ⇒ ∀n m p. SRM r n (SRM r m p) = SRM r (SRM r n m) p

plus_zero_left

|- ∀r. is_semi_ring r ⇒ ∀n. SRP r (SR0 r) n = n

mult_one_left

|- ∀r. is_semi_ring r ⇒ ∀n. SRM r (SR1 r) n = n

mult_zero_left

|- ∀r. is_semi_ring r ⇒ ∀n. SRM r (SR0 r) n = SR0 r

distr_left

|- ∀r.
     is_semi_ring r ⇒
     ∀n m p. SRM r (SRP r n m) p = SRP r (SRM r n p) (SRM r m p)

plus_zero_right

|- ∀r. is_semi_ring r ⇒ ∀n. SRP r n (SR0 r) = n

mult_one_right

|- ∀r. is_semi_ring r ⇒ ∀n. SRM r n (SR1 r) = n

mult_zero_right

|- ∀r. is_semi_ring r ⇒ ∀n. SRM r n (SR0 r) = SR0 r

distr_right

|- ∀r.
     is_semi_ring r ⇒
     ∀m n p. SRM r m (SRP r n p) = SRP r (SRM r m n) (SRM r m p)

plus_rotate

|- ∀r. is_semi_ring r ⇒ ∀m n p. SRP r (SRP r m n) p = SRP r (SRP r n p) m

plus_permute

|- ∀r. is_semi_ring r ⇒ ∀m n p. SRP r (SRP r m n) p = SRP r (SRP r m p) n

mult_rotate

|- ∀r. is_semi_ring r ⇒ ∀m n p. SRM r (SRM r m n) p = SRM r (SRM r n p) m

mult_permute

|- ∀r. is_semi_ring r ⇒ ∀m n p. SRM r (SRM r m n) p = SRM r (SRM r m p) n