Structure semi_ringTheory
signature semi_ringTheory =
sig
type thm = Thm.thm
(* Definitions *)
val is_semi_ring_def : thm
val semi_ring_SR0 : thm
val semi_ring_SR0_fupd : thm
val semi_ring_SR1 : thm
val semi_ring_SR1_fupd : thm
val semi_ring_SRM : thm
val semi_ring_SRM_fupd : thm
val semi_ring_SRP : thm
val semi_ring_SRP_fupd : thm
val semi_ring_TY_DEF : thm
val semi_ring_case_def : thm
val semi_ring_size_def : thm
(* Theorems *)
val EXISTS_semi_ring : thm
val FORALL_semi_ring : thm
val datatype_semi_ring : thm
val distr_left : thm
val distr_right : thm
val mult_assoc : thm
val mult_one_left : thm
val mult_one_right : thm
val mult_permute : thm
val mult_rotate : thm
val mult_sym : thm
val mult_zero_left : thm
val mult_zero_right : thm
val plus_assoc : thm
val plus_permute : thm
val plus_rotate : thm
val plus_sym : thm
val plus_zero_left : thm
val plus_zero_right : thm
val semi_ring_11 : thm
val semi_ring_Axiom : thm
val semi_ring_accessors : thm
val semi_ring_accfupds : thm
val semi_ring_case_cong : thm
val semi_ring_component_equality : thm
val semi_ring_fn_updates : thm
val semi_ring_fupdcanon : thm
val semi_ring_fupdcanon_comp : thm
val semi_ring_fupdfupds : thm
val semi_ring_fupdfupds_comp : thm
val semi_ring_induction : thm
val semi_ring_literal_11 : thm
val semi_ring_literal_nchotomy : thm
val semi_ring_nchotomy : thm
val semi_ring_updates_eq_literal : thm
val semi_ring_grammars : type_grammar.grammar * term_grammar.grammar
val IMPORT : abstraction.inst_infos ->
{ is_semi_ring_def : thm,
semi_ring_SRM_fupd : thm,
semi_ring_SRP_fupd : thm,
semi_ring_SR1_fupd : thm,
semi_ring_SR0_fupd : thm,
semi_ring_SRM : thm,
semi_ring_SRP : thm,
semi_ring_SR1 : thm,
semi_ring_SR0 : thm,
semi_ring_size_def : thm,
semi_ring_case_def : thm,
semi_ring_TY_DEF : thm,
mult_permute : thm,
mult_rotate : thm,
plus_permute : thm,
plus_rotate : thm,
distr_right : thm,
mult_zero_right : thm,
mult_one_right : thm,
plus_zero_right : thm,
distr_left : thm,
mult_zero_left : thm,
mult_one_left : thm,
plus_zero_left : thm,
mult_assoc : thm,
mult_sym : thm,
plus_assoc : thm,
plus_sym : thm,
semi_ring_induction : thm,
semi_ring_Axiom : thm,
semi_ring_nchotomy : thm,
semi_ring_case_cong : thm,
semi_ring_11 : thm,
datatype_semi_ring : thm,
semi_ring_literal_11 : thm,
EXISTS_semi_ring : thm,
FORALL_semi_ring : thm,
semi_ring_literal_nchotomy : thm,
semi_ring_updates_eq_literal : thm,
semi_ring_component_equality : thm,
semi_ring_fupdcanon_comp : thm,
semi_ring_fupdcanon : thm,
semi_ring_fupdfupds_comp : thm,
semi_ring_fupdfupds : thm,
semi_ring_accfupds : thm,
semi_ring_fn_updates : thm,
semi_ring_accessors : thm }
(*
[quantHeuristics] Parent theory of "semi_ring"
[rich_list] Parent theory of "semi_ring"
[is_semi_ring_def] Definition
|- ∀r.
is_semi_ring r ⇔
(∀n m. r.SRP n m = r.SRP m n) ∧
(∀n m p. r.SRP n (r.SRP m p) = r.SRP (r.SRP n m) p) ∧
(∀n m. r.SRM n m = r.SRM m n) ∧
(∀n m p. r.SRM n (r.SRM m p) = r.SRM (r.SRM n m) p) ∧
(∀n. r.SRP r.SR0 n = n) ∧ (∀n. r.SRM r.SR1 n = n) ∧
(∀n. r.SRM r.SR0 n = r.SR0) ∧
∀n m p. r.SRM (r.SRP n m) p = r.SRP (r.SRM n p) (r.SRM m p)
[semi_ring_SR0] Definition
|- ∀a a0 f f0. (semi_ring a a0 f f0).SR0 = a
[semi_ring_SR0_fupd] Definition
|- ∀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] Definition
|- ∀a a0 f f0. (semi_ring a a0 f f0).SR1 = a0
[semi_ring_SR1_fupd] Definition
|- ∀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_SRM] Definition
|- ∀a a0 f f0. (semi_ring a a0 f f0).SRM = f0
[semi_ring_SRM_fupd] Definition
|- ∀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_SRP] Definition
|- ∀a a0 f f0. (semi_ring a a0 f f0).SRP = f
[semi_ring_SRP_fupd] Definition
|- ∀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_TY_DEF] Definition
|- ∃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] Definition
|- ∀a0 a1 a2 a3 f.
semi_ring_CASE (semi_ring a0 a1 a2 a3) f = f a0 a1 a2 a3
[semi_ring_size_def] Definition
|- ∀f a0 a1 a2 a3.
semi_ring_size f (semi_ring a0 a1 a2 a3) = 1 + (f a0 + f a1)
[EXISTS_semi_ring] Theorem
|- ∀P.
(∃s. P s) ⇔
∃a0 a f0 f. P <|SR0 := a0; SR1 := a; SRP := f0; SRM := f|>
[FORALL_semi_ring] Theorem
|- ∀P.
(∀s. P s) ⇔
∀a0 a f0 f. P <|SR0 := a0; SR1 := a; SRP := f0; SRM := f|>
[datatype_semi_ring] Theorem
|- DATATYPE (record semi_ring SR0 SR1 SRP SRM)
[distr_left] Theorem
|- ∀r.
is_semi_ring r ⇒
∀n m p. r.SRM (r.SRP n m) p = r.SRP (r.SRM n p) (r.SRM m p)
[distr_right] Theorem
|- ∀r.
is_semi_ring r ⇒
∀m n p. r.SRM m (r.SRP n p) = r.SRP (r.SRM m n) (r.SRM m p)
[mult_assoc] Theorem
|- ∀r.
is_semi_ring r ⇒
∀n m p. r.SRM n (r.SRM m p) = r.SRM (r.SRM n m) p
[mult_one_left] Theorem
|- ∀r. is_semi_ring r ⇒ ∀n. r.SRM r.SR1 n = n
[mult_one_right] Theorem
|- ∀r. is_semi_ring r ⇒ ∀n. r.SRM n r.SR1 = n
[mult_permute] Theorem
|- ∀r.
is_semi_ring r ⇒
∀m n p. r.SRM (r.SRM m n) p = r.SRM (r.SRM m p) n
[mult_rotate] Theorem
|- ∀r.
is_semi_ring r ⇒
∀m n p. r.SRM (r.SRM m n) p = r.SRM (r.SRM n p) m
[mult_sym] Theorem
|- ∀r. is_semi_ring r ⇒ ∀n m. r.SRM n m = r.SRM m n
[mult_zero_left] Theorem
|- ∀r. is_semi_ring r ⇒ ∀n. r.SRM r.SR0 n = r.SR0
[mult_zero_right] Theorem
|- ∀r. is_semi_ring r ⇒ ∀n. r.SRM n r.SR0 = r.SR0
[plus_assoc] Theorem
|- ∀r.
is_semi_ring r ⇒
∀n m p. r.SRP n (r.SRP m p) = r.SRP (r.SRP n m) p
[plus_permute] Theorem
|- ∀r.
is_semi_ring r ⇒
∀m n p. r.SRP (r.SRP m n) p = r.SRP (r.SRP m p) n
[plus_rotate] Theorem
|- ∀r.
is_semi_ring r ⇒
∀m n p. r.SRP (r.SRP m n) p = r.SRP (r.SRP n p) m
[plus_sym] Theorem
|- ∀r. is_semi_ring r ⇒ ∀n m. r.SRP n m = r.SRP m n
[plus_zero_left] Theorem
|- ∀r. is_semi_ring r ⇒ ∀n. r.SRP r.SR0 n = n
[plus_zero_right] Theorem
|- ∀r. is_semi_ring r ⇒ ∀n. r.SRP n r.SR0 = n
[semi_ring_11] Theorem
|- ∀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_Axiom] Theorem
|- ∀f. ∃fn. ∀a0 a1 a2 a3. fn (semi_ring a0 a1 a2 a3) = f a0 a1 a2 a3
[semi_ring_accessors] Theorem
|- (∀a a0 f f0. (semi_ring a a0 f f0).SR0 = a) ∧
(∀a a0 f f0. (semi_ring a a0 f f0).SR1 = a0) ∧
(∀a a0 f f0. (semi_ring a a0 f f0).SRP = f) ∧
∀a a0 f f0. (semi_ring a a0 f f0).SRM = f0
[semi_ring_accfupds] Theorem
|- (∀s f. (s with SR1 updated_by f).SR0 = s.SR0) ∧
(∀s f. (s with SRP updated_by f).SR0 = s.SR0) ∧
(∀s f. (s with SRM updated_by f).SR0 = s.SR0) ∧
(∀s f. (s with SR0 updated_by f).SR1 = s.SR1) ∧
(∀s f. (s with SRP updated_by f).SR1 = s.SR1) ∧
(∀s f. (s with SRM updated_by f).SR1 = s.SR1) ∧
(∀s f. (s with SR0 updated_by f).SRP = s.SRP) ∧
(∀s f. (s with SR1 updated_by f).SRP = s.SRP) ∧
(∀s f. (s with SRM updated_by f).SRP = s.SRP) ∧
(∀s f. (s with SR0 updated_by f).SRM = s.SRM) ∧
(∀s f. (s with SR1 updated_by f).SRM = s.SRM) ∧
(∀s f. (s with SRP updated_by f).SRM = s.SRM) ∧
(∀s f. (s with SR0 updated_by f).SR0 = f s.SR0) ∧
(∀s f. (s with SR1 updated_by f).SR1 = f s.SR1) ∧
(∀s f. (s with SRP updated_by f).SRP = f s.SRP) ∧
∀s f. (s with SRM updated_by f).SRM = f s.SRM
[semi_ring_case_cong] Theorem
|- ∀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_component_equality] Theorem
|- ∀s1 s2.
(s1 = s2) ⇔
(s1.SR0 = s2.SR0) ∧ (s1.SR1 = s2.SR1) ∧ (s1.SRP = s2.SRP) ∧
(s1.SRM = s2.SRM)
[semi_ring_fn_updates] Theorem
|- (∀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_fupdcanon] Theorem
|- (∀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] Theorem
|- ((∀g f. SR1_fupd f o SR0_fupd g = SR0_fupd g o SR1_fupd f) ∧
∀h g f.
SR1_fupd f o SR0_fupd g o h = SR0_fupd g o SR1_fupd f o h) ∧
((∀g f. SRP_fupd f o SR0_fupd g = SR0_fupd g o SRP_fupd f) ∧
∀h g f.
SRP_fupd f o SR0_fupd g o h = SR0_fupd g o SRP_fupd f o h) ∧
((∀g f. SRP_fupd f o SR1_fupd g = SR1_fupd g o SRP_fupd f) ∧
∀h g f.
SRP_fupd f o SR1_fupd g o h = SR1_fupd g o SRP_fupd f o h) ∧
((∀g f. SRM_fupd f o SR0_fupd g = SR0_fupd g o SRM_fupd f) ∧
∀h g f.
SRM_fupd f o SR0_fupd g o h = SR0_fupd g o SRM_fupd f o h) ∧
((∀g f. SRM_fupd f o SR1_fupd g = SR1_fupd g o SRM_fupd f) ∧
∀h g f.
SRM_fupd f o SR1_fupd g o h = SR1_fupd g o SRM_fupd f o h) ∧
(∀g f. SRM_fupd f o SRP_fupd g = SRP_fupd g o SRM_fupd f) ∧
∀h g f. SRM_fupd f o SRP_fupd g o h = SRP_fupd g o SRM_fupd f o h
[semi_ring_fupdfupds] Theorem
|- (∀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] Theorem
|- ((∀g f. SR0_fupd f o SR0_fupd g = SR0_fupd (f o g)) ∧
∀h g f. SR0_fupd f o SR0_fupd g o h = SR0_fupd (f o g) o h) ∧
((∀g f. SR1_fupd f o SR1_fupd g = SR1_fupd (f o g)) ∧
∀h g f. SR1_fupd f o SR1_fupd g o h = SR1_fupd (f o g) o h) ∧
((∀g f. SRP_fupd f o SRP_fupd g = SRP_fupd (f o g)) ∧
∀h g f. SRP_fupd f o SRP_fupd g o h = SRP_fupd (f o g) o h) ∧
(∀g f. SRM_fupd f o SRM_fupd g = SRM_fupd (f o g)) ∧
∀h g f. SRM_fupd f o SRM_fupd g o h = SRM_fupd (f o g) o h
[semi_ring_induction] Theorem
|- ∀P. (∀a a0 f f0. P (semi_ring a a0 f f0)) ⇒ ∀s. P s
[semi_ring_literal_11] Theorem
|- ∀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)
[semi_ring_literal_nchotomy] Theorem
|- ∀s. ∃a0 a f0 f. s = <|SR0 := a0; SR1 := a; SRP := f0; SRM := f|>
[semi_ring_nchotomy] Theorem
|- ∀ss. ∃a a0 f f0. ss = semi_ring a a0 f f0
[semi_ring_updates_eq_literal] Theorem
|- ∀s a0 a f0 f.
s with <|SR0 := a0; SR1 := a; SRP := f0; SRM := f|> =
<|SR0 := a0; SR1 := a; SRP := f0; SRM := f|>
*)
end
HOL 4, Kananaskis-11