Structure ringTheory
signature ringTheory =
sig
type thm = Thm.thm
(* Definitions *)
val is_ring_def : thm
val ring_R0 : thm
val ring_R0_fupd : thm
val ring_R1 : thm
val ring_R1_fupd : thm
val ring_RM : thm
val ring_RM_fupd : thm
val ring_RN : thm
val ring_RN_fupd : thm
val ring_RP : thm
val ring_RP_fupd : thm
val ring_TY_DEF : thm
val ring_case_def : thm
val ring_size_def : thm
val semi_ring_of_def : thm
(* Theorems *)
val EXISTS_ring : thm
val FORALL_ring : thm
val datatype_ring : thm
val distr_left : thm
val mult_assoc : thm
val mult_one_left : thm
val mult_one_right : thm
val mult_sym : thm
val mult_zero_left : thm
val mult_zero_right : thm
val neg_mult : thm
val opp_def : thm
val plus_assoc : thm
val plus_sym : thm
val plus_zero_left : thm
val plus_zero_right : thm
val ring_11 : thm
val ring_Axiom : thm
val ring_accessors : thm
val ring_accfupds : thm
val ring_case_cong : thm
val ring_component_equality : thm
val ring_fn_updates : thm
val ring_fupdcanon : thm
val ring_fupdcanon_comp : thm
val ring_fupdfupds : thm
val ring_fupdfupds_comp : thm
val ring_induction : thm
val ring_is_semi_ring : thm
val ring_literal_11 : thm
val ring_literal_nchotomy : thm
val ring_nchotomy : thm
val ring_updates_eq_literal : thm
val ring_grammars : type_grammar.grammar * term_grammar.grammar
val IMPORT : abstraction.inst_infos ->
{ semi_ring_of_def : thm,
is_ring_def : thm,
ring_RN_fupd : thm,
ring_RM_fupd : thm,
ring_RP_fupd : thm,
ring_R1_fupd : thm,
ring_R0_fupd : thm,
ring_RN : thm,
ring_RM : thm,
ring_RP : thm,
ring_R1 : thm,
ring_R0 : thm,
ring_size_def : thm,
ring_case_def : thm,
ring_TY_DEF : thm,
neg_mult : thm,
mult_one_right : thm,
ring_is_semi_ring : thm,
mult_zero_right : thm,
mult_zero_left : thm,
plus_zero_right : thm,
distr_left : thm,
opp_def : thm,
mult_one_left : thm,
plus_zero_left : thm,
mult_assoc : thm,
mult_sym : thm,
plus_assoc : thm,
plus_sym : thm,
ring_induction : thm,
ring_Axiom : thm,
ring_nchotomy : thm,
ring_case_cong : thm,
ring_11 : thm,
datatype_ring : thm,
ring_literal_11 : thm,
EXISTS_ring : thm,
FORALL_ring : thm,
ring_literal_nchotomy : thm,
ring_updates_eq_literal : thm,
ring_component_equality : thm,
ring_fupdcanon_comp : thm,
ring_fupdcanon : thm,
ring_fupdfupds_comp : thm,
ring_fupdfupds : thm,
ring_accfupds : thm,
ring_fn_updates : thm,
ring_accessors : thm }
(*
[semi_ring] Parent theory of "ring"
[is_ring_def] Definition
|- ∀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)
[ring_R0] Definition
|- ∀a a0 f f0 f1. R0 (ring a a0 f f0 f1) = a
[ring_R0_fupd] Definition
|- ∀f2 a a0 f f0 f1.
ring a a0 f f0 f1 with R0 updated_by f2 = ring (f2 a) a0 f f0 f1
[ring_R1] Definition
|- ∀a a0 f f0 f1. R1 (ring a a0 f f0 f1) = a0
[ring_R1_fupd] Definition
|- ∀f2 a a0 f f0 f1.
ring a a0 f f0 f1 with R1 updated_by f2 = ring a (f2 a0) f f0 f1
[ring_RM] Definition
|- ∀a a0 f f0 f1. RM (ring a a0 f f0 f1) = f0
[ring_RM_fupd] Definition
|- ∀f2 a a0 f f0 f1.
ring a a0 f f0 f1 with RM updated_by f2 = ring a a0 f (f2 f0) f1
[ring_RN] Definition
|- ∀a a0 f f0 f1. RN (ring a a0 f f0 f1) = f1
[ring_RN_fupd] Definition
|- ∀f2 a a0 f f0 f1.
ring a a0 f f0 f1 with RN updated_by f2 = ring a a0 f f0 (f2 f1)
[ring_RP] Definition
|- ∀a a0 f f0 f1. RP (ring a a0 f f0 f1) = f
[ring_RP_fupd] Definition
|- ∀f2 a a0 f f0 f1.
ring a a0 f f0 f1 with RP updated_by f2 = ring a a0 (f2 f) f0 f1
[ring_TY_DEF] Definition
|- ∃rep.
TYPE_DEFINITION
(λa0'.
∀'ring' .
(∀a0'.
(∃a0 a1 a2 a3 a4.
a0' =
(λa0 a1 a2 a3 a4.
ind_type$CONSTR 0 (a0,a1,a2,a3,a4)
(λn. ind_type$BOTTOM)) a0 a1 a2 a3 a4) ⇒
'ring' a0') ⇒
'ring' a0') rep
[ring_case_def] Definition
|- ∀a0 a1 a2 a3 a4 f.
ring_CASE (ring a0 a1 a2 a3 a4) f = f a0 a1 a2 a3 a4
[ring_size_def] Definition
|- ∀f a0 a1 a2 a3 a4.
ring_size f (ring a0 a1 a2 a3 a4) = 1 + (f a0 + f a1)
[semi_ring_of_def] Definition
|- ∀r. semi_ring_of r = semi_ring (R0 r) (R1 r) (RP r) (RM r)
[EXISTS_ring] Theorem
|- ∀P.
(∃r. P r) ⇔
∃a0 a f1 f0 f.
P <|R0 := a0; R1 := a; RP := f1; RM := f0; RN := f|>
[FORALL_ring] Theorem
|- ∀P.
(∀r. P r) ⇔
∀a0 a f1 f0 f.
P <|R0 := a0; R1 := a; RP := f1; RM := f0; RN := f|>
[datatype_ring] Theorem
|- DATATYPE (record ring R0 R1 RP RM RN)
[distr_left] Theorem
|- ∀r.
is_ring r ⇒
∀n m p. RM r (RP r n m) p = RP r (RM r n p) (RM r m p)
[mult_assoc] Theorem
|- ∀r. is_ring r ⇒ ∀n m p. RM r n (RM r m p) = RM r (RM r n m) p
[mult_one_left] Theorem
|- ∀r. is_ring r ⇒ ∀n. RM r (R1 r) n = n
[mult_one_right] Theorem
|- ∀r. is_ring r ⇒ ∀n. RM r n (R1 r) = n
[mult_sym] Theorem
|- ∀r. is_ring r ⇒ ∀n m. RM r n m = RM r m n
[mult_zero_left] Theorem
|- ∀r. is_ring r ⇒ ∀n. RM r (R0 r) n = R0 r
[mult_zero_right] Theorem
|- ∀r. is_ring r ⇒ ∀n. RM r n (R0 r) = R0 r
[neg_mult] Theorem
|- ∀r. is_ring r ⇒ ∀a b. RM r (RN r a) b = RN r (RM r a b)
[opp_def] Theorem
|- ∀r. is_ring r ⇒ ∀n. RP r n (RN r n) = R0 r
[plus_assoc] Theorem
|- ∀r. is_ring r ⇒ ∀n m p. RP r n (RP r m p) = RP r (RP r n m) p
[plus_sym] Theorem
|- ∀r. is_ring r ⇒ ∀n m. RP r n m = RP r m n
[plus_zero_left] Theorem
|- ∀r. is_ring r ⇒ ∀n. RP r (R0 r) n = n
[plus_zero_right] Theorem
|- ∀r. is_ring r ⇒ ∀n. RP r n (R0 r) = n
[ring_11] Theorem
|- ∀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_Axiom] Theorem
|- ∀f.
∃fn.
∀a0 a1 a2 a3 a4. fn (ring a0 a1 a2 a3 a4) = f a0 a1 a2 a3 a4
[ring_accessors] Theorem
|- (∀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_accfupds] Theorem
|- (∀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_case_cong] Theorem
|- ∀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)) ⇒
(ring_CASE M f = ring_CASE M' f')
[ring_component_equality] Theorem
|- ∀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_fn_updates] Theorem
|- (∀f2 a a0 f f0 f1.
ring a a0 f f0 f1 with R0 updated_by f2 =
ring (f2 a) a0 f f0 f1) ∧
(∀f2 a a0 f f0 f1.
ring a a0 f f0 f1 with R1 updated_by f2 =
ring a (f2 a0) f f0 f1) ∧
(∀f2 a a0 f f0 f1.
ring a a0 f f0 f1 with RP updated_by f2 =
ring a a0 (f2 f) f0 f1) ∧
(∀f2 a a0 f f0 f1.
ring a a0 f f0 f1 with RM updated_by f2 =
ring a a0 f (f2 f0) f1) ∧
∀f2 a a0 f f0 f1.
ring a a0 f f0 f1 with RN updated_by f2 = ring a a0 f f0 (f2 f1)
[ring_fupdcanon] Theorem
|- (∀r g f.
r with <|R1 updated_by f; R0 updated_by g|> =
r with <|R0 updated_by g; R1 updated_by f|>) ∧
(∀r g f.
r with <|RP updated_by f; R0 updated_by g|> =
r with <|R0 updated_by g; RP updated_by f|>) ∧
(∀r g f.
r with <|RP updated_by f; R1 updated_by g|> =
r with <|R1 updated_by g; RP updated_by f|>) ∧
(∀r g f.
r with <|RM updated_by f; R0 updated_by g|> =
r with <|R0 updated_by g; RM updated_by f|>) ∧
(∀r g f.
r with <|RM updated_by f; R1 updated_by g|> =
r with <|R1 updated_by g; RM updated_by f|>) ∧
(∀r g f.
r with <|RM updated_by f; RP updated_by g|> =
r with <|RP updated_by g; RM updated_by f|>) ∧
(∀r g f.
r with <|RN updated_by f; R0 updated_by g|> =
r with <|R0 updated_by g; RN updated_by f|>) ∧
(∀r g f.
r with <|RN updated_by f; R1 updated_by g|> =
r with <|R1 updated_by g; RN updated_by f|>) ∧
(∀r g f.
r with <|RN updated_by f; RP updated_by g|> =
r with <|RP updated_by g; RN updated_by f|>) ∧
∀r g f.
r with <|RN updated_by f; RM updated_by g|> =
r with <|RM updated_by g; RN updated_by f|>
[ring_fupdcanon_comp] Theorem
|- ((∀g f.
_ record fupdateR1 f o _ record fupdateR0 g =
_ record fupdateR0 g o _ record fupdateR1 f) ∧
∀h g f.
_ record fupdateR1 f o _ record fupdateR0 g o h =
_ record fupdateR0 g o _ record fupdateR1 f o h) ∧
((∀g f.
_ record fupdateRP f o _ record fupdateR0 g =
_ record fupdateR0 g o _ record fupdateRP f) ∧
∀h g f.
_ record fupdateRP f o _ record fupdateR0 g o h =
_ record fupdateR0 g o _ record fupdateRP f o h) ∧
((∀g f.
_ record fupdateRP f o _ record fupdateR1 g =
_ record fupdateR1 g o _ record fupdateRP f) ∧
∀h g f.
_ record fupdateRP f o _ record fupdateR1 g o h =
_ record fupdateR1 g o _ record fupdateRP f o h) ∧
((∀g f.
_ record fupdateRM f o _ record fupdateR0 g =
_ record fupdateR0 g o _ record fupdateRM f) ∧
∀h g f.
_ record fupdateRM f o _ record fupdateR0 g o h =
_ record fupdateR0 g o _ record fupdateRM f o h) ∧
((∀g f.
_ record fupdateRM f o _ record fupdateR1 g =
_ record fupdateR1 g o _ record fupdateRM f) ∧
∀h g f.
_ record fupdateRM f o _ record fupdateR1 g o h =
_ record fupdateR1 g o _ record fupdateRM f o h) ∧
((∀g f.
_ record fupdateRM f o _ record fupdateRP g =
_ record fupdateRP g o _ record fupdateRM f) ∧
∀h g f.
_ record fupdateRM f o _ record fupdateRP g o h =
_ record fupdateRP g o _ record fupdateRM f o h) ∧
((∀g f.
_ record fupdateRN f o _ record fupdateR0 g =
_ record fupdateR0 g o _ record fupdateRN f) ∧
∀h g f.
_ record fupdateRN f o _ record fupdateR0 g o h =
_ record fupdateR0 g o _ record fupdateRN f o h) ∧
((∀g f.
_ record fupdateRN f o _ record fupdateR1 g =
_ record fupdateR1 g o _ record fupdateRN f) ∧
∀h g f.
_ record fupdateRN f o _ record fupdateR1 g o h =
_ record fupdateR1 g o _ record fupdateRN f o h) ∧
((∀g f.
_ record fupdateRN f o _ record fupdateRP g =
_ record fupdateRP g o _ record fupdateRN f) ∧
∀h g f.
_ record fupdateRN f o _ record fupdateRP g o h =
_ record fupdateRP g o _ record fupdateRN f o h) ∧
(∀g f.
_ record fupdateRN f o _ record fupdateRM g =
_ record fupdateRM g o _ record fupdateRN f) ∧
∀h g f.
_ record fupdateRN f o _ record fupdateRM g o h =
_ record fupdateRM g o _ record fupdateRN f o h
[ring_fupdfupds] Theorem
|- (∀r g f.
r with <|R0 updated_by f; R0 updated_by g|> =
r with R0 updated_by f o g) ∧
(∀r g f.
r with <|R1 updated_by f; R1 updated_by g|> =
r with R1 updated_by f o g) ∧
(∀r g f.
r with <|RP updated_by f; RP updated_by g|> =
r with RP updated_by f o g) ∧
(∀r g f.
r with <|RM updated_by f; RM updated_by g|> =
r with RM updated_by f o g) ∧
∀r g f.
r with <|RN updated_by f; RN updated_by g|> =
r with RN updated_by f o g
[ring_fupdfupds_comp] Theorem
|- ((∀g f.
_ record fupdateR0 f o _ record fupdateR0 g =
_ record fupdateR0 (f o g)) ∧
∀h g f.
_ record fupdateR0 f o _ record fupdateR0 g o h =
_ record fupdateR0 (f o g) o h) ∧
((∀g f.
_ record fupdateR1 f o _ record fupdateR1 g =
_ record fupdateR1 (f o g)) ∧
∀h g f.
_ record fupdateR1 f o _ record fupdateR1 g o h =
_ record fupdateR1 (f o g) o h) ∧
((∀g f.
_ record fupdateRP f o _ record fupdateRP g =
_ record fupdateRP (f o g)) ∧
∀h g f.
_ record fupdateRP f o _ record fupdateRP g o h =
_ record fupdateRP (f o g) o h) ∧
((∀g f.
_ record fupdateRM f o _ record fupdateRM g =
_ record fupdateRM (f o g)) ∧
∀h g f.
_ record fupdateRM f o _ record fupdateRM g o h =
_ record fupdateRM (f o g) o h) ∧
(∀g f.
_ record fupdateRN f o _ record fupdateRN g =
_ record fupdateRN (f o g)) ∧
∀h g f.
_ record fupdateRN f o _ record fupdateRN g o h =
_ record fupdateRN (f o g) o h
[ring_induction] Theorem
|- ∀P. (∀a a0 f f0 f1. P (ring a a0 f f0 f1)) ⇒ ∀r. P r
[ring_is_semi_ring] Theorem
|- ∀r. is_ring r ⇒ is_semi_ring (semi_ring_of r)
[ring_literal_11] Theorem
|- ∀a01 a1 f11 f01 f1 a02 a2 f12 f02 f2.
(<|R0 := a01; R1 := a1; RP := f11; RM := f01; RN := f1|> =
<|R0 := a02; R1 := a2; RP := f12; RM := f02; RN := f2|>) ⇔
(a01 = a02) ∧ (a1 = a2) ∧ (f11 = f12) ∧ (f01 = f02) ∧ (f1 = f2)
[ring_literal_nchotomy] Theorem
|- ∀r.
∃a0 a f1 f0 f.
r = <|R0 := a0; R1 := a; RP := f1; RM := f0; RN := f|>
[ring_nchotomy] Theorem
|- ∀rr. ∃a a0 f f0 f1. rr = ring a a0 f f0 f1
[ring_updates_eq_literal] Theorem
|- ∀r a0 a f1 f0 f.
r with <|R0 := a0; R1 := a; RP := f1; RM := f0; RN := f|> =
<|R0 := a0; R1 := a; RP := f1; RM := f0; RN := f|>
*)
end
HOL 4, Kananaskis-10