- ring_TY_DEF
-
|- ?rep.
TYPE_DEFINITION
(\a0'.
!'ring'.
(!a0'.
(?a0 a1 a2 a3 a4.
a0' =
(\a0 a1 a2 a3 a4. CONSTR 0 (a0,a1,a2,a3,a4) (\n. BOTTOM)) a0
a1 a2 a3 a4) ==>
'ring' a0') ==>
'ring' a0') rep
- ring_repfns
-
|- (!a. mk_ring (dest_ring a) = a) /\
!r.
(\a0'.
!'ring'.
(!a0'.
(?a0 a1 a2 a3 a4.
a0' =
(\a0 a1 a2 a3 a4. CONSTR 0 (a0,a1,a2,a3,a4) (\n. BOTTOM)) a0
a1 a2 a3 a4) ==>
'ring' a0') ==>
'ring' a0') r =
(dest_ring (mk_ring r) = r)
- ring0_def
-
|- ring0 =
(\a0 a1 a2 a3 a4.
mk_ring
((\a0 a1 a2 a3 a4. CONSTR 0 (a0,a1,a2,a3,a4) (\n. BOTTOM)) a0 a1 a2 a3
a4))
- ring
-
|- ring = ring0
- ring_case_def
-
|- !f a0 a1 a2 a3 a4. case f (ring a0 a1 a2 a3 a4) = f a0 a1 a2 a3 a4
- ring_size_def
-
|- !f a0 a1 a2 a3 a4. ring_size f (ring a0 a1 a2 a3 a4) = 1 + (f a0 + f a1)
- ring_R0
-
|- !a a0 f f0 f1. R0 (ring a a0 f f0 f1) = a
- ring_R1
-
|- !a a0 f f0 f1. R1 (ring a a0 f f0 f1) = a0
- ring_RP
-
|- !a a0 f f0 f1. RP (ring a a0 f f0 f1) = f
- ring_RM
-
|- !a a0 f f0 f1. RM (ring a a0 f f0 f1) = f0
- ring_RN
-
|- !a a0 f f0 f1. RN (ring a a0 f f0 f1) = f1
- ring_R0_update
-
|- !a1 a a0 f f0 f1. ring a a0 f f0 f1 with R0 := a1 = ring a1 a0 f f0 f1
- ring_R1_update
-
|- !a1 a a0 f f0 f1. ring a a0 f f0 f1 with R1 := a1 = ring a a1 f f0 f1
- ring_RP_update
-
|- !f2 a a0 f f0 f1. ring a a0 f f0 f1 with RP := f2 = ring a a0 f2 f0 f1
- ring_RM_update
-
|- !f2 a a0 f f0 f1. ring a a0 f f0 f1 with RM := f2 = ring a a0 f f2 f1
- ring_RN_update
-
|- !f2 a a0 f f0 f1. ring a a0 f f0 f1 with RN := f2 = ring a a0 f f0 f2
- ring_R0_fupd
-
|- !f x. x with R0 updated_by f = x with R0 := f (R0 x)
- ring_R1_fupd
-
|- !f x. x with R1 updated_by f = x with R1 := f (R1 x)
- ring_RP_fupd
-
|- !f x. x with RP updated_by f = x with RP := f (RP x)
- ring_RM_fupd
-
|- !f x. x with RM updated_by f = x with RM := f (RM x)
- ring_RN_fupd
-
|- !f x. x with RN updated_by f = x with RN := f (RN x)
- is_ring_def
-
|- !r.
is_ring r =
(!n m. RP r n m = RP r m n) /\
(!n m p. RP r n (RP r m p) = RP r (RP r n m) p) /\
(!n m. RM r n m = RM r m n) /\
(!n m p. RM r n (RM r m p) = RM r (RM r n m) p) /\
(!n. RP r (R0 r) n = n) /\ (!n. RM r (R1 r) n = n) /\
(!n. RP r n (RN r n) = R0 r) /\
!n m p. RM r (RP r n m) p = RP r (RM r n p) (RM r m p)
- semi_ring_of_def
-
|- !r. semi_ring_of r = semi_ring (R0 r) (R1 r) (RP r) (RM r)
- ring_accessors
-
|- (!a a0 f f0 f1. R0 (ring a a0 f f0 f1) = a) /\
(!a a0 f f0 f1. R1 (ring a a0 f f0 f1) = a0) /\
(!a a0 f f0 f1. RP (ring a a0 f f0 f1) = f) /\
(!a a0 f f0 f1. RM (ring a a0 f f0 f1) = f0) /\
!a a0 f f0 f1. RN (ring a a0 f f0 f1) = f1
- ring_updates
-
|- (!a1 a a0 f f0 f1. ring a a0 f f0 f1 with R0 := a1 = ring a1 a0 f f0 f1) /\
(!a1 a a0 f f0 f1. ring a a0 f f0 f1 with R1 := a1 = ring a a1 f f0 f1) /\
(!f2 a a0 f f0 f1. ring a a0 f f0 f1 with RP := f2 = ring a a0 f2 f0 f1) /\
(!f2 a a0 f f0 f1. ring a a0 f f0 f1 with RM := f2 = ring a a0 f f2 f1) /\
!f2 a a0 f f0 f1. ring a a0 f f0 f1 with RN := f2 = ring a a0 f f0 f2
- ring_fn_updates
-
|- (!f x. x with R0 updated_by f = x with R0 := f (R0 x)) /\
(!f x. x with R1 updated_by f = x with R1 := f (R1 x)) /\
(!f x. x with RP updated_by f = x with RP := f (RP x)) /\
(!f x. x with RM updated_by f = x with RM := f (RM x)) /\
!f x. x with RN updated_by f = x with RN := f (RN x)
- ring_accupds
-
|- (!x r. R0 (r with R1 := x) = R0 r) /\ (!x r. R0 (r with RP := x) = R0 r) /\
(!x r. R0 (r with RM := x) = R0 r) /\ (!x r. R0 (r with RN := x) = R0 r) /\
(!x r. R1 (r with R0 := x) = R1 r) /\ (!x r. R1 (r with RP := x) = R1 r) /\
(!x r. R1 (r with RM := x) = R1 r) /\ (!x r. R1 (r with RN := x) = R1 r) /\
(!x r. RP (r with R0 := x) = RP r) /\ (!x r. RP (r with R1 := x) = RP r) /\
(!x r. RP (r with RM := x) = RP r) /\ (!x r. RP (r with RN := x) = RP r) /\
(!x r. RM (r with R0 := x) = RM r) /\ (!x r. RM (r with R1 := x) = RM r) /\
(!x r. RM (r with RP := x) = RM r) /\ (!x r. RM (r with RN := x) = RM r) /\
(!x r. RN (r with R0 := x) = RN r) /\ (!x r. RN (r with R1 := x) = RN r) /\
(!x r. RN (r with RP := x) = RN r) /\ (!x r. RN (r with RM := x) = RN r) /\
(!x r. R0 (r with R0 := x) = x) /\ (!x r. R1 (r with R1 := x) = x) /\
(!x r. RP (r with RP := x) = x) /\ (!x r. RM (r with RM := x) = x) /\
!x r. RN (r with RN := x) = x
- ring_accfupds
-
|- (!r f. R0 (r with R1 updated_by f) = R0 r) /\
(!r f. R0 (r with RP updated_by f) = R0 r) /\
(!r f. R0 (r with RM updated_by f) = R0 r) /\
(!r f. R0 (r with RN updated_by f) = R0 r) /\
(!r f. R1 (r with R0 updated_by f) = R1 r) /\
(!r f. R1 (r with RP updated_by f) = R1 r) /\
(!r f. R1 (r with RM updated_by f) = R1 r) /\
(!r f. R1 (r with RN updated_by f) = R1 r) /\
(!r f. RP (r with R0 updated_by f) = RP r) /\
(!r f. RP (r with R1 updated_by f) = RP r) /\
(!r f. RP (r with RM updated_by f) = RP r) /\
(!r f. RP (r with RN updated_by f) = RP r) /\
(!r f. RM (r with R0 updated_by f) = RM r) /\
(!r f. RM (r with R1 updated_by f) = RM r) /\
(!r f. RM (r with RP updated_by f) = RM r) /\
(!r f. RM (r with RN updated_by f) = RM r) /\
(!r f. RN (r with R0 updated_by f) = RN r) /\
(!r f. RN (r with R1 updated_by f) = RN r) /\
(!r f. RN (r with RP updated_by f) = RN r) /\
(!r f. RN (r with RM updated_by f) = RN r) /\
(!r f. R0 (r with R0 updated_by f) = f (R0 r)) /\
(!r f. R1 (r with R1 updated_by f) = f (R1 r)) /\
(!r f. RP (r with RP updated_by f) = f (RP r)) /\
(!r f. RM (r with RM updated_by f) = f (RM r)) /\
!r f. RN (r with RN updated_by f) = f (RN r)
- ring_updaccs
-
|- (!r. r with R0 := R0 r = r) /\ (!r. r with R1 := R1 r = r) /\
(!r. r with RP := RP r = r) /\ (!r. r with RM := RM r = r) /\
!r. r with RN := RN r = r
- ring_cupdaccs
-
|- (!val r. (val = R0 r) ==> (r with R0 := val = r)) /\
(!val r. (val = R1 r) ==> (r with R1 := val = r)) /\
(!val r. (val = RP r) ==> (r with RP := val = r)) /\
(!val r. (val = RM r) ==> (r with RM := val = r)) /\
!val r. (val = RN r) ==> (r with RN := val = r)
- ring_updupds
-
|- (!x2 x1 r. r with <|R0 := x1; R0 := x2|> = r with R0 := x1) /\
(!x2 x1 r. r with <|R1 := x1; R1 := x2|> = r with R1 := x1) /\
(!x2 x1 r. r with <|RP := x1; RP := x2|> = r with RP := x1) /\
(!x2 x1 r. r with <|RM := x1; RM := x2|> = r with RM := x1) /\
!x2 x1 r. r with <|RN := x1; RN := x2|> = r with RN := x1
- ring_updcanon
-
|- (!z x r. r with <|R1 := x; R0 := z|> = r with <|R0 := z; R1 := x|>) /\
(!z x r. r with <|RP := x; R0 := z|> = r with <|R0 := z; RP := x|>) /\
(!z x r. r with <|RP := x; R1 := z|> = r with <|R1 := z; RP := x|>) /\
(!z x r. r with <|RM := x; R0 := z|> = r with <|R0 := z; RM := x|>) /\
(!z x r. r with <|RM := x; R1 := z|> = r with <|R1 := z; RM := x|>) /\
(!z x r. r with <|RM := x; RP := z|> = r with <|RP := z; RM := x|>) /\
(!z x r. r with <|RN := x; R0 := z|> = r with <|R0 := z; RN := x|>) /\
(!z x r. r with <|RN := x; R1 := z|> = r with <|R1 := z; RN := x|>) /\
(!z x r. r with <|RN := x; RP := z|> = r with <|RP := z; RN := x|>) /\
!z x r. r with <|RN := x; RM := z|> = r with <|RM := z; RN := x|>
- ring_R0_update_semi11
-
|- !x y r1 r2. (r1 with R0 := x = r2 with R0 := y) ==> (x = y)
- ring_R1_update_semi11
-
|- !x y r1 r2. (r1 with R1 := x = r2 with R1 := y) ==> (x = y)
- ring_RP_update_semi11
-
|- !x y r1 r2. (r1 with RP := x = r2 with RP := y) ==> (x = y)
- ring_RM_update_semi11
-
|- !x y r1 r2. (r1 with RM := x = r2 with RM := y) ==> (x = y)
- ring_RN_update_semi11
-
|- !x y r1 r2. (r1 with RN := x = r2 with RN := y) ==> (x = y)
- ring_component_equality
-
|- !r1 r2.
(r1 = r2) =
(R0 r1 = R0 r2) /\ (R1 r1 = R1 r2) /\ (RP r1 = RP r2) /\
(RM r1 = RM r2) /\ (RN r1 = RN r2)
- ring_updates_eq_literal
-
|- !r f1 f0 f a0 a.
r with <|R0 := a0; R1 := a; RP := f1; RM := f0; RN := f|> =
<|R0 := a0; R1 := a; RP := f1; RM := f0; RN := f|>
- ring_11
-
|- !a0 a1 a2 a3 a4 a0' a1' a2' a3' a4'.
(ring a0 a1 a2 a3 a4 = ring a0' a1' a2' a3' a4') =
(a0 = a0') /\ (a1 = a1') /\ (a2 = a2') /\ (a3 = a3') /\ (a4 = a4')
- ring_case_cong
-
|- !M M' f.
(M = M') /\
(!a0 a1 a2 a3 a4.
(M' = ring a0 a1 a2 a3 a4) ==>
(f a0 a1 a2 a3 a4 = f' a0 a1 a2 a3 a4)) ==>
(case f M = case f' M')
- ring_nchotomy
-
|- !r. ?a a0 f f0 f1. r = ring a a0 f f0 f1
- ring_Axiom
-
|- !f. ?fn. !a0 a1 a2 a3 a4. fn (ring a0 a1 a2 a3 a4) = f a0 a1 a2 a3 a4
- ring_induction
-
|- !P. (!a a0 f f0 f1. P (ring a a0 f f0 f1)) ==> !r. P r
- plus_sym
-
|- !r. is_ring r ==> !n m. RP r n m = RP r m n
- plus_assoc
-
|- !r. is_ring r ==> !n m p. RP r n (RP r m p) = RP r (RP r n m) p
- mult_sym
-
|- !r. is_ring r ==> !n m. RM r n m = RM r m n
- mult_assoc
-
|- !r. is_ring r ==> !n m p. RM r n (RM r m p) = RM r (RM r n m) p
- plus_zero_left
-
|- !r. is_ring r ==> !n. RP r (R0 r) n = n
- mult_one_left
-
|- !r. is_ring r ==> !n. RM r (R1 r) n = n
- opp_def
-
|- !r. is_ring r ==> !n. RP r n (RN r n) = R0 r
- distr_left
-
|- !r. is_ring r ==> !n m p. RM r (RP r n m) p = RP r (RM r n p) (RM r m p)
- plus_zero_right
-
|- !r. is_ring r ==> !n. RP r n (R0 r) = n
- mult_zero_left
-
|- !r. is_ring r ==> !n. RM r (R0 r) n = R0 r
- mult_zero_right
-
|- !r. is_ring r ==> !n. RM r n (R0 r) = R0 r
- ring_is_semi_ring
-
|- !r. is_ring r ==> is_semi_ring (semi_ring_of r)
- mult_one_right
-
|- !r. is_ring r ==> !n. RM r n (R1 r) = n
- neg_mult
-
|- !r. is_ring r ==> !a b. RM r (RN r a) b = RN r (RM r a b)