- semi_ring_TY_DEF
-
|- ?rep.
TYPE_DEFINITION
(\a0'.
!'semi_ring'.
(!a0'.
(?a0 a1 a2 a3.
a0' =
(\a0 a1 a2 a3. CONSTR 0 (a0,a1,a2,a3) (\n. BOTTOM)) a0 a1 a2
a3) ==>
'semi_ring' a0') ==>
'semi_ring' a0') rep
- semi_ring_repfns
-
|- (!a. mk_semi_ring (dest_semi_ring a) = a) /\
!r.
(\a0'.
!'semi_ring'.
(!a0'.
(?a0 a1 a2 a3.
a0' =
(\a0 a1 a2 a3. CONSTR 0 (a0,a1,a2,a3) (\n. BOTTOM)) a0 a1 a2
a3) ==>
'semi_ring' a0') ==>
'semi_ring' a0') r =
(dest_semi_ring (mk_semi_ring r) = r)
- semi_ring0_def
-
|- semi_ring0 =
(\a0 a1 a2 a3.
mk_semi_ring
((\a0 a1 a2 a3. CONSTR 0 (a0,a1,a2,a3) (\n. BOTTOM)) a0 a1 a2 a3))
- semi_ring
-
|- semi_ring = semi_ring0
- semi_ring_case_def
-
|- !f a0 a1 a2 a3. case f (semi_ring a0 a1 a2 a3) = 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_update
-
|- !a1 a a0 f f0. semi_ring a a0 f f0 with SR0 := a1 = semi_ring a1 a0 f f0
- semi_ring_SR1_update
-
|- !a1 a a0 f f0. semi_ring a a0 f f0 with SR1 := a1 = semi_ring a a1 f f0
- semi_ring_SRP_update
-
|- !f1 a a0 f f0. semi_ring a a0 f f0 with SRP := f1 = semi_ring a a0 f1 f0
- semi_ring_SRM_update
-
|- !f1 a a0 f f0. semi_ring a a0 f f0 with SRM := f1 = semi_ring a a0 f f1
- semi_ring_SR0_fupd
-
|- !f x. x with SR0 updated_by f = x with SR0 := f (SR0 x)
- semi_ring_SR1_fupd
-
|- !f x. x with SR1 updated_by f = x with SR1 := f (SR1 x)
- semi_ring_SRP_fupd
-
|- !f x. x with SRP updated_by f = x with SRP := f (SRP x)
- semi_ring_SRM_fupd
-
|- !f x. x with SRM updated_by f = x with SRM := f (SRM x)
- 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)
- 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_updates
-
|- (!a1 a a0 f f0.
semi_ring a a0 f f0 with SR0 := a1 = semi_ring a1 a0 f f0) /\
(!a1 a a0 f f0.
semi_ring a a0 f f0 with SR1 := a1 = semi_ring a a1 f f0) /\
(!f1 a a0 f f0.
semi_ring a a0 f f0 with SRP := f1 = semi_ring a a0 f1 f0) /\
!f1 a a0 f f0. semi_ring a a0 f f0 with SRM := f1 = semi_ring a a0 f f1
- semi_ring_fn_updates
-
|- (!f x. x with SR0 updated_by f = x with SR0 := f (SR0 x)) /\
(!f x. x with SR1 updated_by f = x with SR1 := f (SR1 x)) /\
(!f x. x with SRP updated_by f = x with SRP := f (SRP x)) /\
!f x. x with SRM updated_by f = x with SRM := f (SRM x)
- semi_ring_accupds
-
|- (!x s. SR0 (s with SR1 := x) = SR0 s) /\
(!x s. SR0 (s with SRP := x) = SR0 s) /\
(!x s. SR0 (s with SRM := x) = SR0 s) /\
(!x s. SR1 (s with SR0 := x) = SR1 s) /\
(!x s. SR1 (s with SRP := x) = SR1 s) /\
(!x s. SR1 (s with SRM := x) = SR1 s) /\
(!x s. SRP (s with SR0 := x) = SRP s) /\
(!x s. SRP (s with SR1 := x) = SRP s) /\
(!x s. SRP (s with SRM := x) = SRP s) /\
(!x s. SRM (s with SR0 := x) = SRM s) /\
(!x s. SRM (s with SR1 := x) = SRM s) /\
(!x s. SRM (s with SRP := x) = SRM s) /\
(!x s. SR0 (s with SR0 := x) = x) /\ (!x s. SR1 (s with SR1 := x) = x) /\
(!x s. SRP (s with SRP := x) = x) /\ !x s. SRM (s with SRM := x) = x
- 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_updaccs
-
|- (!s. s with SR0 := SR0 s = s) /\ (!s. s with SR1 := SR1 s = s) /\
(!s. s with SRP := SRP s = s) /\ !s. s with SRM := SRM s = s
- semi_ring_cupdaccs
-
|- (!val s. (val = SR0 s) ==> (s with SR0 := val = s)) /\
(!val s. (val = SR1 s) ==> (s with SR1 := val = s)) /\
(!val s. (val = SRP s) ==> (s with SRP := val = s)) /\
!val s. (val = SRM s) ==> (s with SRM := val = s)
- semi_ring_updupds
-
|- (!x2 x1 s. s with <|SR0 := x1; SR0 := x2|> = s with SR0 := x1) /\
(!x2 x1 s. s with <|SR1 := x1; SR1 := x2|> = s with SR1 := x1) /\
(!x2 x1 s. s with <|SRP := x1; SRP := x2|> = s with SRP := x1) /\
!x2 x1 s. s with <|SRM := x1; SRM := x2|> = s with SRM := x1
- semi_ring_updcanon
-
|- (!z x s. s with <|SR1 := x; SR0 := z|> = s with <|SR0 := z; SR1 := x|>) /\
(!z x s. s with <|SRP := x; SR0 := z|> = s with <|SR0 := z; SRP := x|>) /\
(!z x s. s with <|SRP := x; SR1 := z|> = s with <|SR1 := z; SRP := x|>) /\
(!z x s. s with <|SRM := x; SR0 := z|> = s with <|SR0 := z; SRM := x|>) /\
(!z x s. s with <|SRM := x; SR1 := z|> = s with <|SR1 := z; SRM := x|>) /\
!z x s. s with <|SRM := x; SRP := z|> = s with <|SRP := z; SRM := x|>
- semi_ring_SR0_update_semi11
-
|- !x y r1 r2. (r1 with SR0 := x = r2 with SR0 := y) ==> (x = y)
- semi_ring_SR1_update_semi11
-
|- !x y r1 r2. (r1 with SR1 := x = r2 with SR1 := y) ==> (x = y)
- semi_ring_SRP_update_semi11
-
|- !x y r1 r2. (r1 with SRP := x = r2 with SRP := y) ==> (x = y)
- semi_ring_SRM_update_semi11
-
|- !x y r1 r2. (r1 with SRM := x = r2 with SRM := y) ==> (x = y)
- 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 f0 f a0 a.
s with <|SR0 := a0; SR1 := a; SRP := f0; SRM := f|> =
<|SR0 := a0; SR1 := a; SRP := f0; SRM := f|>
- 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)) ==>
(case f M = case f' M')
- semi_ring_nchotomy
-
|- !s. ?a a0 f f0. s = 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