Structure sptreeTheory
signature sptreeTheory =
sig
type thm = Thm.thm
(* Definitions *)
val delete_def : thm
val difference_def : thm
val domain_def : thm
val foldi_def : thm
val fromAList_primitive_def : thm
val fromList_def : thm
val insert_curried_def : thm
val insert_tupled_primitive_def : thm
val inter_def : thm
val inter_eq_def : thm
val lookup_curried_def : thm
val lookup_tupled_primitive_def : thm
val lrnext_def : thm
val mk_BN_curried_def : thm
val mk_BN_tupled_primitive_def : thm
val mk_BS_curried_def : thm
val mk_BS_tupled_primitive_def : thm
val mk_wf_def : thm
val size_def : thm
val spt_TY_DEF : thm
val spt_case_def : thm
val spt_size_def : thm
val toAList_def : thm
val toListA_def : thm
val toList_def : thm
val union_def : thm
val wf_def : thm
(* Theorems *)
val ALOOKUP_toAList : thm
val FINITE_domain : thm
val MEM_toAList : thm
val datatype_spt : thm
val delete_compute : thm
val delete_mk_wf : thm
val domain_delete : thm
val domain_empty : thm
val domain_foldi : thm
val domain_fromAList : thm
val domain_fromList : thm
val domain_insert : thm
val domain_inter : thm
val domain_lookup : thm
val domain_mk_wf : thm
val domain_sing : thm
val domain_union : thm
val fromAList_def : thm
val fromAList_ind : thm
val fromAList_toAList : thm
val insert_compute : thm
val insert_def : thm
val insert_ind : thm
val insert_mk_wf : thm
val insert_notEmpty : thm
val insert_union : thm
val inter_LN : thm
val inter_assoc : thm
val inter_eq : thm
val isEmpty_toList : thm
val isEmpty_toListA : thm
val isEmpty_union : thm
val lookup_NONE_domain : thm
val lookup_compute : thm
val lookup_def : thm
val lookup_delete : thm
val lookup_difference : thm
val lookup_fromAList : thm
val lookup_fromAList_toAList : thm
val lookup_fromList : thm
val lookup_ind : thm
val lookup_insert : thm
val lookup_insert1 : thm
val lookup_inter : thm
val lookup_inter_eq : thm
val lookup_mk_wf : thm
val lookup_union : thm
val lrnext_thm : thm
val mk_BN_def : thm
val mk_BN_ind : thm
val mk_BS_def : thm
val mk_BS_ind : thm
val mk_wf_eq : thm
val set_foldi_keys : thm
val spt_11 : thm
val spt_Axiom : thm
val spt_case_cong : thm
val spt_distinct : thm
val spt_eq_thm : thm
val spt_induction : thm
val spt_nchotomy : thm
val toListA_append : thm
val union_LN : thm
val union_assoc : thm
val union_mk_wf : thm
val wf_delete : thm
val wf_fromAList : thm
val wf_insert : thm
val wf_inter : thm
val wf_mk_id : thm
val wf_mk_wf : thm
val wf_union : thm
val sptree_grammars : type_grammar.grammar * term_grammar.grammar
(*
[alist] Parent theory of "sptree"
[logroot] Parent theory of "sptree"
[delete_def] Definition
|- (∀k. isEmpty (delete k LN)) ∧
(∀k a. delete k (LS a) = if k = 0 then LN else LS a) ∧
(∀k t1 t2.
delete k (BN t1 t2) =
if k = 0 then BN t1 t2
else if EVEN k then mk_BN (delete ((k − 1) DIV 2) t1) t2
else mk_BN t1 (delete ((k − 1) DIV 2) t2)) ∧
∀k t1 a t2.
delete k (BS t1 a t2) =
if k = 0 then BN t1 t2
else if EVEN k then mk_BS (delete ((k − 1) DIV 2) t1) a t2
else mk_BS t1 a (delete ((k − 1) DIV 2) t2)
[difference_def] Definition
|- (∀t. isEmpty (difference LN t)) ∧
(∀a t.
difference (LS a) t =
case t of
LN => LS a
| LS b => LN
| BN t1 t2 => LS a
| BS t1' b' t2' => LN) ∧
(∀t1 t2 t.
difference (BN t1 t2) t =
case t of
LN => BN t1 t2
| LS a => BN t1 t2
| BN t1' t2' => mk_BN (difference t1 t1') (difference t2 t2')
| BS t1'' a'' t2'' =>
mk_BN (difference t1 t1'') (difference t2 t2'')) ∧
∀t1 a t2 t.
difference (BS t1 a t2) t =
case t of
LN => BS t1 a t2
| LS a' => BN t1 t2
| BN t1' t2' => mk_BS (difference t1 t1') a (difference t2 t2')
| BS t1'' a''' t2'' =>
mk_BN (difference t1 t1'') (difference t2 t2'')
[domain_def] Definition
|- (domain LN = ∅) ∧ (∀v0. domain (LS v0) = {0}) ∧
(∀t1 t2.
domain (BN t1 t2) =
IMAGE (λn. 2 * n + 2) (domain t1) ∪
IMAGE (λn. 2 * n + 1) (domain t2)) ∧
∀t1 v1 t2.
domain (BS t1 v1 t2) =
{0} ∪ IMAGE (λn. 2 * n + 2) (domain t1) ∪
IMAGE (λn. 2 * n + 1) (domain t2)
[foldi_def] Definition
|- (∀f i acc. foldi f i acc LN = acc) ∧
(∀f i acc a. foldi f i acc (LS a) = f i a acc) ∧
(∀f i acc t1 t2.
foldi f i acc (BN t1 t2) =
(let inc = sptree$lrnext i
in
foldi f (i + inc) (foldi f (i + 2 * inc) acc t1) t2)) ∧
∀f i acc t1 a t2.
foldi f i acc (BS t1 a t2) =
(let inc = sptree$lrnext i
in
foldi f (i + inc) (f i a (foldi f (i + 2 * inc) acc t1)) t2)
[fromAList_primitive_def] Definition
|- fromAList =
WFREC (@R. WF R ∧ ∀y x xs. R xs ((x,y)::xs))
(λfromAList a.
case a of
[] => I LN
| (x,y)::xs => I (insert x y (fromAList xs)))
[fromList_def] Definition
|- ∀l.
fromList l =
SND (FOLDL (λ(i,t) a. (i + 1,insert i a t)) (0,LN) l)
[insert_curried_def] Definition
|- ∀x x1 x2. insert x x1 x2 = insert_tupled (x,x1,x2)
[insert_tupled_primitive_def] Definition
|- insert_tupled =
WFREC
(@R.
WF R ∧
(∀a k. k ≠ 0 ∧ ¬EVEN k ⇒ R ((k − 1) DIV 2,a,LN) (k,a,LN)) ∧
(∀a k. k ≠ 0 ∧ EVEN k ⇒ R ((k − 1) DIV 2,a,LN) (k,a,LN)) ∧
(∀a' a k.
k ≠ 0 ∧ ¬EVEN k ⇒ R ((k − 1) DIV 2,a,LN) (k,a,LS a')) ∧
(∀a' a k.
k ≠ 0 ∧ EVEN k ⇒ R ((k − 1) DIV 2,a,LN) (k,a,LS a')) ∧
(∀t1 t2 a k.
k ≠ 0 ∧ ¬EVEN k ⇒ R ((k − 1) DIV 2,a,t2) (k,a,BN t1 t2)) ∧
(∀t2 t1 a k.
k ≠ 0 ∧ EVEN k ⇒ R ((k − 1) DIV 2,a,t1) (k,a,BN t1 t2)) ∧
(∀t2 a' t1 a k.
k ≠ 0 ∧ EVEN k ⇒
R ((k − 1) DIV 2,a,t1) (k,a,BS t1 a' t2)) ∧
∀a' t1 t2 a k.
k ≠ 0 ∧ ¬EVEN k ⇒ R ((k − 1) DIV 2,a,t2) (k,a,BS t1 a' t2))
(λinsert_tupled a''.
case a'' of
(k,a,LN) =>
I
(if k = 0 then LS a
else if EVEN k then
BN (insert_tupled ((k − 1) DIV 2,a,LN)) LN
else BN LN (insert_tupled ((k − 1) DIV 2,a,LN)))
| (k,a,LS a') =>
I
(if k = 0 then LS a
else if EVEN k then
BS (insert_tupled ((k − 1) DIV 2,a,LN)) a' LN
else BS LN a' (insert_tupled ((k − 1) DIV 2,a,LN)))
| (k,a,BN t1 t2) =>
I
(if k = 0 then BS t1 a t2
else if EVEN k then
BN (insert_tupled ((k − 1) DIV 2,a,t1)) t2
else BN t1 (insert_tupled ((k − 1) DIV 2,a,t2)))
| (k,a,BS t1' a''' t2') =>
I
(if k = 0 then BS t1' a t2'
else if EVEN k then
BS (insert_tupled ((k − 1) DIV 2,a,t1')) a''' t2'
else
BS t1' a''' (insert_tupled ((k − 1) DIV 2,a,t2'))))
[inter_def] Definition
|- (∀t. isEmpty (inter LN t)) ∧
(∀a t.
inter (LS a) t =
case t of
LN => LN
| LS b => LS a
| BN t1 t2 => LN
| BS t1' v4 t2' => LS a) ∧
(∀t1 t2 t.
inter (BN t1 t2) t =
case t of
LN => LN
| LS a => LN
| BN t1' t2' => mk_BN (inter t1 t1') (inter t2 t2')
| BS t1'' a'' t2'' => mk_BN (inter t1 t1'') (inter t2 t2'')) ∧
∀t1 a t2 t.
inter (BS t1 a t2) t =
case t of
LN => LN
| LS a' => LS a
| BN t1' t2' => mk_BN (inter t1 t1') (inter t2 t2')
| BS t1'' a''' t2'' => mk_BS (inter t1 t1'') a (inter t2 t2'')
[inter_eq_def] Definition
|- (∀t. isEmpty (inter_eq LN t)) ∧
(∀a t.
inter_eq (LS a) t =
case t of
LN => LN
| LS b => if a = b then LS a else LN
| BN t1 t2 => LN
| BS t1' b' t2' => if a = b' then LS a else LN) ∧
(∀t1 t2 t.
inter_eq (BN t1 t2) t =
case t of
LN => LN
| LS a => LN
| BN t1' t2' => mk_BN (inter_eq t1 t1') (inter_eq t2 t2')
| BS t1'' a'' t2'' =>
mk_BN (inter_eq t1 t1'') (inter_eq t2 t2'')) ∧
∀t1 a t2 t.
inter_eq (BS t1 a t2) t =
case t of
LN => LN
| LS a' => if a' = a then LS a else LN
| BN t1' t2' => mk_BN (inter_eq t1 t1') (inter_eq t2 t2')
| BS t1'' a''' t2'' =>
if a''' = a then
mk_BS (inter_eq t1 t1'') a (inter_eq t2 t2'')
else mk_BN (inter_eq t1 t1'') (inter_eq t2 t2'')
[lookup_curried_def] Definition
|- ∀x x1. lookup x x1 = lookup_tupled (x,x1)
[lookup_tupled_primitive_def] Definition
|- lookup_tupled =
WFREC
(@R.
WF R ∧
(∀t2 t1 k.
k ≠ 0 ⇒
R ((k − 1) DIV 2,if EVEN k then t1 else t2)
(k,BN t1 t2)) ∧
∀a t2 t1 k.
k ≠ 0 ⇒
R ((k − 1) DIV 2,if EVEN k then t1 else t2) (k,BS t1 a t2))
(λlookup_tupled a'.
case a' of
(k,LN) => I NONE
| (k,LS a) => I (if k = 0 then SOME a else NONE)
| (k,BN t1 t2) =>
I
(if k = 0 then NONE
else
lookup_tupled
((k − 1) DIV 2,if EVEN k then t1 else t2))
| (k,BS t1' a'' t2') =>
I
(if k = 0 then SOME a''
else
lookup_tupled
((k − 1) DIV 2,if EVEN k then t1' else t2')))
[lrnext_def] Definition
|- (sptree$lrnext ZERO = 1) ∧
(∀n. sptree$lrnext (BIT1 n) = 2 * sptree$lrnext n) ∧
∀n. sptree$lrnext (BIT2 n) = 2 * sptree$lrnext n
[mk_BN_curried_def] Definition
|- ∀x x1. mk_BN x x1 = mk_BN_tupled (x,x1)
[mk_BN_tupled_primitive_def] Definition
|- mk_BN_tupled =
WFREC (@R. WF R)
(λmk_BN_tupled a.
case a of
(LN,LN) => I LN
| (LN,LS v20) => I (BN LN (LS v20))
| (LN,BN v21 v22) => I (BN LN (BN v21 v22))
| (LN,BS v23 v24 v25) => I (BN LN (BS v23 v24 v25))
| (LS v8,v1) => I (BN (LS v8) v1)
| (BN v9 v10,v1) => I (BN (BN v9 v10) v1)
| (BS v11 v12 v13,v1) => I (BN (BS v11 v12 v13) v1))
[mk_BS_curried_def] Definition
|- ∀x x1 x2. mk_BS x x1 x2 = mk_BS_tupled (x,x1,x2)
[mk_BS_tupled_primitive_def] Definition
|- mk_BS_tupled =
WFREC (@R. WF R)
(λmk_BS_tupled a.
case a of
(LN,x,LN) => I (LS x)
| (LS v22,x,LN) => I (BS (LS v22) x LN)
| (BN v23 v24,x,LN) => I (BS (BN v23 v24) x LN)
| (BS v25 v26 v27,x,LN) => I (BS (BS v25 v26 v27) x LN)
| (v,x,LS v10) => I (BS v x (LS v10))
| (v,x,BN v11 v12) => I (BS v x (BN v11 v12))
| (v,x,BS v13 v14 v15) => I (BS v x (BS v13 v14 v15)))
[mk_wf_def] Definition
|- isEmpty (mk_wf LN) ∧ (∀x. mk_wf (LS x) = LS x) ∧
(∀t1 t2. mk_wf (BN t1 t2) = mk_BN (mk_wf t1) (mk_wf t2)) ∧
∀t1 x t2. mk_wf (BS t1 x t2) = mk_BS (mk_wf t1) x (mk_wf t2)
[size_def] Definition
|- (size LN = 0) ∧ (∀a. size (LS a) = 1) ∧
(∀t1 t2. size (BN t1 t2) = size t1 + size t2) ∧
∀t1 a t2. size (BS t1 a t2) = size t1 + size t2 + 1
[spt_TY_DEF] Definition
|- ∃rep.
TYPE_DEFINITION
(λa0'.
∀'spt' .
(∀a0'.
(a0' = ind_type$CONSTR 0 ARB (λn. ind_type$BOTTOM)) ∨
(∃a.
a0' =
(λa.
ind_type$CONSTR (SUC 0) a (λn. ind_type$BOTTOM))
a) ∨
(∃a0 a1.
(a0' =
(λa0 a1.
ind_type$CONSTR (SUC (SUC 0)) ARB
(ind_type$FCONS a0
(ind_type$FCONS a1
(λn. ind_type$BOTTOM)))) a0 a1) ∧
'spt' a0 ∧ 'spt' a1) ∨
(∃a0 a1 a2.
(a0' =
(λa0 a1 a2.
ind_type$CONSTR (SUC (SUC (SUC 0))) a1
(ind_type$FCONS a0
(ind_type$FCONS a2
(λn. ind_type$BOTTOM)))) a0 a1 a2) ∧
'spt' a0 ∧ 'spt' a2) ⇒
'spt' a0') ⇒
'spt' a0') rep
[spt_case_def] Definition
|- (∀v f f1 f2. spt_CASE LN v f f1 f2 = v) ∧
(∀a v f f1 f2. spt_CASE (LS a) v f f1 f2 = f a) ∧
(∀a0 a1 v f f1 f2. spt_CASE (BN a0 a1) v f f1 f2 = f1 a0 a1) ∧
∀a0 a1 a2 v f f1 f2.
spt_CASE (BS a0 a1 a2) v f f1 f2 = f2 a0 a1 a2
[spt_size_def] Definition
|- (∀f. spt_size f LN = 0) ∧ (∀f a. spt_size f (LS a) = 1 + f a) ∧
(∀f a0 a1.
spt_size f (BN a0 a1) = 1 + (spt_size f a0 + spt_size f a1)) ∧
∀f a0 a1 a2.
spt_size f (BS a0 a1 a2) =
1 + (spt_size f a0 + (f a1 + spt_size f a2))
[toAList_def] Definition
|- toAList = foldi (λk v a. (k,v)::a) 0 []
[toListA_def] Definition
|- (∀acc. toListA acc LN = acc) ∧
(∀acc a. toListA acc (LS a) = a::acc) ∧
(∀acc t1 t2.
toListA acc (BN t1 t2) = toListA (toListA acc t2) t1) ∧
∀acc t1 a t2.
toListA acc (BS t1 a t2) = toListA (a::toListA acc t2) t1
[toList_def] Definition
|- ∀m. toList m = toListA [] m
[union_def] Definition
|- (∀t. union LN t = t) ∧
(∀a t.
union (LS a) t =
case t of
LN => LS a
| LS b => LS a
| BN t1 t2 => BS t1 a t2
| BS t1' v4 t2' => BS t1' a t2') ∧
(∀t1 t2 t.
union (BN t1 t2) t =
case t of
LN => BN t1 t2
| LS a => BS t1 a t2
| BN t1' t2' => BN (union t1 t1') (union t2 t2')
| BS t1'' a'' t2'' => BS (union t1 t1'') a'' (union t2 t2'')) ∧
∀t1 a t2 t.
union (BS t1 a t2) t =
case t of
LN => BS t1 a t2
| LS a' => BS t1 a t2
| BN t1' t2' => BS (union t1 t1') a (union t2 t2')
| BS t1'' a''' t2'' => BS (union t1 t1'') a (union t2 t2'')
[wf_def] Definition
|- (wf LN ⇔ T) ∧ (∀a. wf (LS a) ⇔ T) ∧
(∀t1 t2.
wf (BN t1 t2) ⇔ wf t1 ∧ wf t2 ∧ ¬(isEmpty t1 ∧ isEmpty t2)) ∧
∀t1 a t2.
wf (BS t1 a t2) ⇔ wf t1 ∧ wf t2 ∧ ¬(isEmpty t1 ∧ isEmpty t2)
[ALOOKUP_toAList] Theorem
|- ∀t x. ALOOKUP (toAList t) x = lookup x t
[FINITE_domain] Theorem
|- FINITE (domain t)
[MEM_toAList] Theorem
|- ∀t k v. MEM (k,v) (toAList t) ⇔ (lookup k t = SOME v)
[datatype_spt] Theorem
|- DATATYPE (spt LN LS BN BS)
[delete_compute] Theorem
|- (delete (NUMERAL n) t = delete n t) ∧ isEmpty (delete 0 LN) ∧
isEmpty (delete 0 (LS a)) ∧ (delete 0 (BN t1 t2) = BN t1 t2) ∧
(delete 0 (BS t1 a t2) = BN t1 t2) ∧ isEmpty (delete ZERO LN) ∧
isEmpty (delete ZERO (LS a)) ∧
(delete ZERO (BN t1 t2) = BN t1 t2) ∧
(delete ZERO (BS t1 a t2) = BN t1 t2) ∧
isEmpty (delete (BIT1 n) LN) ∧ (delete (BIT1 n) (LS a) = LS a) ∧
(delete (BIT1 n) (BN t1 t2) = mk_BN t1 (delete n t2)) ∧
(delete (BIT1 n) (BS t1 a t2) = mk_BS t1 a (delete n t2)) ∧
isEmpty (delete (BIT2 n) LN) ∧ (delete (BIT2 n) (LS a) = LS a) ∧
(delete (BIT2 n) (BN t1 t2) = mk_BN (delete n t1) t2) ∧
(delete (BIT2 n) (BS t1 a t2) = mk_BS (delete n t1) a t2)
[delete_mk_wf] Theorem
|- ∀x t. delete x (mk_wf t) = mk_wf (delete x t)
[domain_delete] Theorem
|- domain (delete k t) = domain t DELETE k
[domain_empty] Theorem
|- ∀t. wf t ⇒ (isEmpty t ⇔ (domain t = ∅))
[domain_foldi] Theorem
|- domain t = foldi (λk v a. k INSERT a) 0 ∅ t
[domain_fromAList] Theorem
|- ∀ls. domain (fromAList ls) = set (MAP FST ls)
[domain_fromList] Theorem
|- domain (fromList l) = count (LENGTH l)
[domain_insert] Theorem
|- domain (insert k v t) = k INSERT domain t
[domain_inter] Theorem
|- domain (inter t1 t2) = domain t1 ∩ domain t2
[domain_lookup] Theorem
|- ∀t k. k ∈ domain t ⇔ ∃v. lookup k t = SOME v
[domain_mk_wf] Theorem
|- ∀t. domain (mk_wf t) = domain t
[domain_sing] Theorem
|- domain (insert k v LN) = {k}
[domain_union] Theorem
|- domain (union t1 t2) = domain t1 ∪ domain t2
[fromAList_def] Theorem
|- isEmpty (fromAList []) ∧
∀y xs x. fromAList ((x,y)::xs) = insert x y (fromAList xs)
[fromAList_ind] Theorem
|- ∀P. P [] ∧ (∀x y xs. P xs ⇒ P ((x,y)::xs)) ⇒ ∀v. P v
[fromAList_toAList] Theorem
|- ∀t. wf t ⇒ (fromAList (toAList t) = t)
[insert_compute] Theorem
|- (insert (NUMERAL n) a t = insert n a t) ∧ (insert 0 a LN = LS a) ∧
(insert 0 a (LS a') = LS a) ∧
(insert 0 a (BN t1 t2) = BS t1 a t2) ∧
(insert 0 a (BS t1 a' t2) = BS t1 a t2) ∧
(insert ZERO a LN = LS a) ∧ (insert ZERO a (LS a') = LS a) ∧
(insert ZERO a (BN t1 t2) = BS t1 a t2) ∧
(insert ZERO a (BS t1 a' t2) = BS t1 a t2) ∧
(insert (BIT1 n) a LN = BN LN (insert n a LN)) ∧
(insert (BIT1 n) a (LS a') = BS LN a' (insert n a LN)) ∧
(insert (BIT1 n) a (BN t1 t2) = BN t1 (insert n a t2)) ∧
(insert (BIT1 n) a (BS t1 a' t2) = BS t1 a' (insert n a t2)) ∧
(insert (BIT2 n) a LN = BN (insert n a LN) LN) ∧
(insert (BIT2 n) a (LS a') = BS (insert n a LN) a' LN) ∧
(insert (BIT2 n) a (BN t1 t2) = BN (insert n a t1) t2) ∧
(insert (BIT2 n) a (BS t1 a' t2) = BS (insert n a t1) a' t2)
[insert_def] Theorem
|- (∀k a.
insert k a LN =
if k = 0 then LS a
else if EVEN k then BN (insert ((k − 1) DIV 2) a LN) LN
else BN LN (insert ((k − 1) DIV 2) a LN)) ∧
(∀k a' a.
insert k a (LS a') =
if k = 0 then LS a
else if EVEN k then BS (insert ((k − 1) DIV 2) a LN) a' LN
else BS LN a' (insert ((k − 1) DIV 2) a LN)) ∧
(∀t2 t1 k a.
insert k a (BN t1 t2) =
if k = 0 then BS t1 a t2
else if EVEN k then BN (insert ((k − 1) DIV 2) a t1) t2
else BN t1 (insert ((k − 1) DIV 2) a t2)) ∧
∀t2 t1 k a' a.
insert k a (BS t1 a' t2) =
if k = 0 then BS t1 a t2
else if EVEN k then BS (insert ((k − 1) DIV 2) a t1) a' t2
else BS t1 a' (insert ((k − 1) DIV 2) a t2)
[insert_ind] Theorem
|- ∀P.
(∀k a.
(k ≠ 0 ∧ EVEN k ⇒ P ((k − 1) DIV 2) a LN) ∧
(k ≠ 0 ∧ ¬EVEN k ⇒ P ((k − 1) DIV 2) a LN) ⇒
P k a LN) ∧
(∀k a a'.
(k ≠ 0 ∧ EVEN k ⇒ P ((k − 1) DIV 2) a LN) ∧
(k ≠ 0 ∧ ¬EVEN k ⇒ P ((k − 1) DIV 2) a LN) ⇒
P k a (LS a')) ∧
(∀k a t1 t2.
(k ≠ 0 ∧ EVEN k ⇒ P ((k − 1) DIV 2) a t1) ∧
(k ≠ 0 ∧ ¬EVEN k ⇒ P ((k − 1) DIV 2) a t2) ⇒
P k a (BN t1 t2)) ∧
(∀k a t1 a' t2.
(k ≠ 0 ∧ EVEN k ⇒ P ((k − 1) DIV 2) a t1) ∧
(k ≠ 0 ∧ ¬EVEN k ⇒ P ((k − 1) DIV 2) a t2) ⇒
P k a (BS t1 a' t2)) ⇒
∀v v1 v2. P v v1 v2
[insert_mk_wf] Theorem
|- ∀x v t. insert x v (mk_wf t) = mk_wf (insert x v t)
[insert_notEmpty] Theorem
|- insert k a t ≠ LN
[insert_union] Theorem
|- ∀k v s. insert k v s = union (insert k v LN) s
[inter_LN] Theorem
|- ∀t. isEmpty (inter t LN) ∧ isEmpty (inter LN t)
[inter_assoc] Theorem
|- ∀t1 t2 t3. inter t1 (inter t2 t3) = inter (inter t1 t2) t3
[inter_eq] Theorem
|- ∀t1 t2 t3 t4.
(inter t1 t2 = inter t3 t4) ⇔
∀x. lookup x (inter t1 t2) = lookup x (inter t3 t4)
[isEmpty_toList] Theorem
|- ∀t. wf t ⇒ (isEmpty t ⇔ (toList t = []))
[isEmpty_toListA] Theorem
|- ∀t acc. wf t ⇒ (isEmpty t ⇔ (toListA acc t = acc))
[isEmpty_union] Theorem
|- isEmpty (union m1 m2) ⇔ isEmpty m1 ∧ isEmpty m2
[lookup_NONE_domain] Theorem
|- (lookup k t = NONE) ⇔ k ∉ domain t
[lookup_compute] Theorem
|- (lookup (NUMERAL n) t = lookup n t) ∧ (lookup 0 LN = NONE) ∧
(lookup 0 (LS a) = SOME a) ∧ (lookup 0 (BN t1 t2) = NONE) ∧
(lookup 0 (BS t1 a t2) = SOME a) ∧ (lookup ZERO LN = NONE) ∧
(lookup ZERO (LS a) = SOME a) ∧ (lookup ZERO (BN t1 t2) = NONE) ∧
(lookup ZERO (BS t1 a t2) = SOME a) ∧
(lookup (BIT1 n) LN = NONE) ∧ (lookup (BIT1 n) (LS a) = NONE) ∧
(lookup (BIT1 n) (BN t1 t2) = lookup n t2) ∧
(lookup (BIT1 n) (BS t1 a t2) = lookup n t2) ∧
(lookup (BIT2 n) LN = NONE) ∧ (lookup (BIT2 n) (LS a) = NONE) ∧
(lookup (BIT2 n) (BN t1 t2) = lookup n t1) ∧
(lookup (BIT2 n) (BS t1 a t2) = lookup n t1)
[lookup_def] Theorem
|- (∀k. lookup k LN = NONE) ∧
(∀k a. lookup k (LS a) = if k = 0 then SOME a else NONE) ∧
(∀t2 t1 k.
lookup k (BN t1 t2) =
if k = 0 then NONE
else lookup ((k − 1) DIV 2) (if EVEN k then t1 else t2)) ∧
∀t2 t1 k a.
lookup k (BS t1 a t2) =
if k = 0 then SOME a
else lookup ((k − 1) DIV 2) (if EVEN k then t1 else t2)
[lookup_delete] Theorem
|- ∀t k1 k2.
lookup k1 (delete k2 t) = if k1 = k2 then NONE else lookup k1 t
[lookup_difference] Theorem
|- ∀m1 m2 k.
lookup k (difference m1 m2) =
if lookup k m2 = NONE then lookup k m1 else NONE
[lookup_fromAList] Theorem
|- ∀ls x. lookup x (fromAList ls) = ALOOKUP ls x
[lookup_fromAList_toAList] Theorem
|- ∀t x. lookup x (fromAList (toAList t)) = lookup x t
[lookup_fromList] Theorem
|- lookup n (fromList l) =
if n < LENGTH l then SOME (EL n l) else NONE
[lookup_ind] Theorem
|- ∀P.
(∀k. P k LN) ∧ (∀k a. P k (LS a)) ∧
(∀k t1 t2.
(k ≠ 0 ⇒ P ((k − 1) DIV 2) (if EVEN k then t1 else t2)) ⇒
P k (BN t1 t2)) ∧
(∀k t1 a t2.
(k ≠ 0 ⇒ P ((k − 1) DIV 2) (if EVEN k then t1 else t2)) ⇒
P k (BS t1 a t2)) ⇒
∀v v1. P v v1
[lookup_insert] Theorem
|- ∀k2 v t k1.
lookup k1 (insert k2 v t) =
if k1 = k2 then SOME v else lookup k1 t
[lookup_insert1] Theorem
|- ∀k a t. lookup k (insert k a t) = SOME a
[lookup_inter] Theorem
|- ∀m1 m2 k.
lookup k (inter m1 m2) =
case (lookup k m1,lookup k m2) of
(SOME v,SOME w) => SOME v
| _ => NONE
[lookup_inter_eq] Theorem
|- ∀m1 m2 k.
lookup k (inter_eq m1 m2) =
case lookup k m1 of
NONE => NONE
| SOME v => if lookup k m2 = SOME v then SOME v else NONE
[lookup_mk_wf] Theorem
|- ∀x t. lookup x (mk_wf t) = lookup x t
[lookup_union] Theorem
|- ∀m1 m2 k.
lookup k (union m1 m2) =
case lookup k m1 of NONE => lookup k m2 | SOME v => SOME v
[lrnext_thm] Theorem
|- (∀a. sptree$lrnext 0 = 1) ∧
(∀n a. sptree$lrnext (NUMERAL n) = sptree$lrnext n) ∧
(sptree$lrnext ZERO = 1) ∧
(∀n. sptree$lrnext (BIT1 n) = 2 * sptree$lrnext n) ∧
∀n. sptree$lrnext (BIT2 n) = 2 * sptree$lrnext n
[mk_BN_def] Theorem
|- isEmpty (mk_BN LN LN) ∧ (mk_BN LN (LS v14) = BN LN (LS v14)) ∧
(mk_BN LN (BN v15 v16) = BN LN (BN v15 v16)) ∧
(mk_BN LN (BS v17 v18 v19) = BN LN (BS v17 v18 v19)) ∧
(mk_BN (LS v2) t2 = BN (LS v2) t2) ∧
(mk_BN (BN v3 v4) t2 = BN (BN v3 v4) t2) ∧
(mk_BN (BS v5 v6 v7) t2 = BN (BS v5 v6 v7) t2)
[mk_BN_ind] Theorem
|- ∀P.
P LN LN ∧ (∀v14. P LN (LS v14)) ∧
(∀v15 v16. P LN (BN v15 v16)) ∧
(∀v17 v18 v19. P LN (BS v17 v18 v19)) ∧ (∀v2 t2. P (LS v2) t2) ∧
(∀v3 v4 t2. P (BN v3 v4) t2) ∧
(∀v5 v6 v7 t2. P (BS v5 v6 v7) t2) ⇒
∀v v1. P v v1
[mk_BS_def] Theorem
|- (mk_BS LN x LN = LS x) ∧
(mk_BS (LS v16) x LN = BS (LS v16) x LN) ∧
(mk_BS (BN v17 v18) x LN = BS (BN v17 v18) x LN) ∧
(mk_BS (BS v19 v20 v21) x LN = BS (BS v19 v20 v21) x LN) ∧
(mk_BS t1 x (LS v4) = BS t1 x (LS v4)) ∧
(mk_BS t1 x (BN v5 v6) = BS t1 x (BN v5 v6)) ∧
(mk_BS t1 x (BS v7 v8 v9) = BS t1 x (BS v7 v8 v9))
[mk_BS_ind] Theorem
|- ∀P.
(∀x. P LN x LN) ∧ (∀v16 x. P (LS v16) x LN) ∧
(∀v17 v18 x. P (BN v17 v18) x LN) ∧
(∀v19 v20 v21 x. P (BS v19 v20 v21) x LN) ∧
(∀t1 x v4. P t1 x (LS v4)) ∧ (∀t1 x v5 v6. P t1 x (BN v5 v6)) ∧
(∀t1 x v7 v8 v9. P t1 x (BS v7 v8 v9)) ⇒
∀v v1 v2. P v v1 v2
[mk_wf_eq] Theorem
|- ∀t1 t2. (mk_wf t1 = mk_wf t2) ⇔ ∀x. lookup x t1 = lookup x t2
[set_foldi_keys] Theorem
|- ∀t a i.
foldi (λk v a. k INSERT a) i a t =
a ∪ IMAGE (λn. i + sptree$lrnext i * n) (domain t)
[spt_11] Theorem
|- (∀a a'. (LS a = LS a') ⇔ (a = a')) ∧
(∀a0 a1 a0' a1'.
(BN a0 a1 = BN a0' a1') ⇔ (a0 = a0') ∧ (a1 = a1')) ∧
∀a0 a1 a2 a0' a1' a2'.
(BS a0 a1 a2 = BS a0' a1' a2') ⇔
(a0 = a0') ∧ (a1 = a1') ∧ (a2 = a2')
[spt_Axiom] Theorem
|- ∀f0 f1 f2 f3.
∃fn.
(fn LN = f0) ∧ (∀a. fn (LS a) = f1 a) ∧
(∀a0 a1. fn (BN a0 a1) = f2 a0 a1 (fn a0) (fn a1)) ∧
∀a0 a1 a2. fn (BS a0 a1 a2) = f3 a1 a0 a2 (fn a0) (fn a2)
[spt_case_cong] Theorem
|- ∀M M' v f f1 f2.
(M = M') ∧ (isEmpty M' ⇒ (v = v')) ∧
(∀a. (M' = LS a) ⇒ (f a = f' a)) ∧
(∀a0 a1. (M' = BN a0 a1) ⇒ (f1 a0 a1 = f1' a0 a1)) ∧
(∀a0 a1 a2. (M' = BS a0 a1 a2) ⇒ (f2 a0 a1 a2 = f2' a0 a1 a2)) ⇒
(spt_CASE M v f f1 f2 = spt_CASE M' v' f' f1' f2')
[spt_distinct] Theorem
|- (∀a. LN ≠ LS a) ∧ (∀a1 a0. LN ≠ BN a0 a1) ∧
(∀a2 a1 a0. LN ≠ BS a0 a1 a2) ∧ (∀a1 a0 a. LS a ≠ BN a0 a1) ∧
(∀a2 a1 a0 a. LS a ≠ BS a0 a1 a2) ∧
∀a2 a1' a1 a0' a0. BN a0 a1 ≠ BS a0' a1' a2
[spt_eq_thm] Theorem
|- ∀t1 t2.
wf t1 ∧ wf t2 ⇒ ((t1 = t2) ⇔ ∀n. lookup n t1 = lookup n t2)
[spt_induction] Theorem
|- ∀P.
P LN ∧ (∀a. P (LS a)) ∧ (∀s s0. P s ∧ P s0 ⇒ P (BN s s0)) ∧
(∀s s0. P s ∧ P s0 ⇒ ∀a. P (BS s a s0)) ⇒
∀s. P s
[spt_nchotomy] Theorem
|- ∀ss.
isEmpty ss ∨ (∃a. ss = LS a) ∨ (∃s s0. ss = BN s s0) ∨
∃s a s0. ss = BS s a s0
[toListA_append] Theorem
|- ∀t acc. toListA acc t = toListA [] t ++ acc
[union_LN] Theorem
|- ∀t. (union t LN = t) ∧ (union LN t = t)
[union_assoc] Theorem
|- ∀t1 t2 t3. union t1 (union t2 t3) = union (union t1 t2) t3
[union_mk_wf] Theorem
|- ∀t1 t2. inter (mk_wf t1) (mk_wf t2) = mk_wf (inter t1 t2)
[wf_delete] Theorem
|- ∀t k. wf t ⇒ wf (delete k t)
[wf_fromAList] Theorem
|- ∀ls. wf (fromAList ls)
[wf_insert] Theorem
|- ∀k a t. wf t ⇒ wf (insert k a t)
[wf_inter] Theorem
|- ∀m1 m2. wf (inter m1 m2)
[wf_mk_id] Theorem
|- ∀t. wf t ⇒ (mk_wf t = t)
[wf_mk_wf] Theorem
|- ∀t. wf (mk_wf t)
[wf_union] Theorem
|- ∀m1 m2. wf m1 ∧ wf m2 ⇒ wf (union m1 m2)
*)
end
HOL 4, Kananaskis-10