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HashtableProofScript.sml
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(*
Proof of the hashtable module
*)
open preamble ml_translatorTheory ml_translatorLib cfLib
mlbasicsProgTheory ArrayProgTheory ArrayProofTheory ListProgTheory
MapProgTheory HashtableProgTheory
comparisonTheory;
val _ = new_theory"HashtableProof";
val _ = translation_extends "HashtableProg";
val hashtable_st = get_ml_prog_state();
(* ----------------------------------- *)
(* the union of each bucket is the heap *)
(* the vlv list contains the buckets *)
(* each bucket only contains keys that hash there *)
Definition hash_key_set_def:
hash_key_set hf length idx = { k' | hf k' MOD length = idx }
End
Definition bucket_ok_def:
bucket_ok b hf idx length = !k v.
(mlmap$lookup b k = SOME v ==> k ∈ (hash_key_set hf length idx))
End
Definition buckets_empty_def:
buckets_empty bs = (MAP mlmap$to_fmap bs = (REPLICATE (LENGTH bs) FEMPTY))
End
Definition buckets_to_fmap_def:
buckets_to_fmap xs = alist_to_fmap (FLAT (MAP mlmap$toAscList xs))
End
Definition list_union_def:
list_union [x] = x /\
list_union (x::xs) = mlmap$union x (list_union xs)
End
Definition buckets_ok_def:
buckets_ok bs hf =
!i. i < LENGTH bs ==>
bucket_ok (EL i bs) hf i (LENGTH bs)
End
Definition hashtable_inv_def:
hashtable_inv a b hf cmp (h:('a|->'b)) vlv =
?buckets.
h = to_fmap (list_union buckets) /\
buckets_ok buckets hf /\
0 <> (LENGTH vlv) /\
LIST_REL (MAP_TYPE a b) buckets vlv /\
EVERY mlmap$map_ok buckets /\
EVERY (\t. mlmap$cmp_of t = cmp) buckets
End
Definition REF_NUM_def:
REF_NUM loc n =
SEP_EXISTS v. REF loc v * & (NUM n v)
End
Definition REF_ARRAY_def:
REF_ARRAY loc arr content = REF loc arr * ARRAY arr content
End
Definition HASHTABLE_def:
HASHTABLE a b hf cmp h v =
SEP_EXISTS ur ar hfv vlv arr cmpv heuristic_size.
&(v = (Conv hashtable_con_stamp [ur; ar; hfv; cmpv]) /\
(a --> NUM) hf hfv /\
(a --> a --> ORDERING_TYPE) cmp cmpv /\
TotOrd cmp /\
hashtable_inv a b hf cmp h vlv) *
REF_NUM ur heuristic_size *
REF_ARRAY ar arr vlv
End
Theorem hashtable_initBuckets_spec:
!a b n nv cmp cmpv.
(a --> a --> ORDERING_TYPE) cmp cmpv /\
NUM n nv ==>
app (p:'ffi ffi_proj) Hashtable_initBuckets_v [nv; cmpv]
emp
(POSTv ar. SEP_EXISTS mpv. &(MAP_TYPE a b (mlmap$empty cmp) mpv) *
ARRAY ar (REPLICATE n mpv))
Proof
rpt strip_tac
\\ xcf_with_def Hashtable_initBuckets_v_def
\\ xlet `POSTv r1. & (MAP_TYPE a b (mlmap$empty cmp) r1)`
>-(xapp
\\ simp[])
\\ xapp_spec array_alloc_spec
\\ xsimpl
\\ asm_exists_tac
\\ simp[]
\\ asm_exists_tac
\\ simp[]
QED
Theorem buckets_ok_empty:
!n cmp hf. TotOrd cmp ==>
buckets_ok (REPLICATE n (mlmap$empty cmp)) hf
Proof
rpt strip_tac
\\ fs [EL_REPLICATE,TotOrder_imp_good_cmp,buckets_ok_def, bucket_ok_def,
mlmapTheory.empty_thm,balanced_mapTheory.empty_thm,
mlmapTheory.lookup_thm, balanced_mapTheory.lookup_thm,flookup_thm]
QED
Theorem list_union_cmp_of:
!cmp h t.
EVERY (\t. cmp_of t = cmp) (h::t) /\
EVERY map_ok (h::t) ==>
cmp_of (list_union (h::t)) = cmp
Proof
rpt strip_tac
\\ Induct_on `(h::t)`
>-(simp[])
>-(rpt strip_tac
\\Cases_on `t`
>-(fs[list_union_def])
>-(fs[list_union_def]
\\ Cases_on `h'`
\\ Cases_on `list_union (h''::t')`
\\ fs[EVERY_DEF, mlmapTheory.union_def, mlmapTheory.cmp_of_def]))
QED
Theorem list_union_map_ok:
!cmp h t.
EVERY (\t. cmp_of t = cmp) (h::t) /\
EVERY map_ok (h::t) ==>
map_ok (list_union (h::t))
Proof
rpt strip_tac
\\ Induct_on `(h::t)`
>-(simp[])
>-(rpt strip_tac
\\ Cases_on `t = []`
\\ rw[]
\\ fs[EVERY_DEF, list_union_def]
>-(Cases_on `t`
>-(fs[])
>-(simp[list_union_def]
\\imp_res_tac list_union_cmp_of
\\rfs[]
\\imp_res_tac mlmapTheory.union_thm
\\rfs[])))
QED
Theorem list_union_thm:
!cmp.
EVERY (\t. cmp_of t = cmp) (h::t) /\
EVERY map_ok (h::t) ==>
map_ok (list_union (h::t)) /\
cmp_of (list_union (h::t)) = cmp /\
(t = [] ==> to_fmap (list_union (h::t)) = to_fmap h ) /\
(t <> [] ==> to_fmap (list_union (h::t)) = to_fmap h ⊌ to_fmap (list_union t))
Proof
rpt strip_tac
>-(fs[EVERY_DEF,list_union_map_ok])
>-(fs[EVERY_DEF, list_union_cmp_of])
>-(rw[]
\\ fs[list_union_def])
\\ Cases_on `t`
>-(fs[])
>-(simp[list_union_def]
\\ `EVERY map_ok (h'::t')` by (imp_res_tac EVERY_DEF \\ rfs[])
\\ `EVERY (\t. cmp_of t = cmp) (h'::t')` by (imp_res_tac EVERY_DEF \\ rfs[])
\\ imp_res_tac list_union_cmp_of
\\ imp_res_tac list_union_map_ok
\\ imp_res_tac mlmapTheory.union_thm
\\ rfs[])
QED
Theorem list_union_empty_maps:
!cmp.
EVERY ($=(mlmap$empty cmp)) (h::t) /\
EVERY (\t. cmp_of t = cmp) (h::t) /\
EVERY map_ok (h::t) ==>
mlmap$to_fmap (list_union (h::t)) = FEMPTY
Proof
rpt strip_tac
\\ Induct_on `(h::t)`
>-(simp[])
>-(rpt strip_tac
\\ Cases_on `t=[]`
>-(rw[]
\\ fs[EVERY_DEF, list_union_def]
\\ ntac 2 (pop_assum kall_tac)
\\ rveq
\\ simp[mlmapTheory.empty_thm])
>-(imp_res_tac list_union_thm
\\rfs[]
\\Cases_on `t`
>-(fs[])
>-(rfs[]
\\ fs[EVERY_DEF]
\\ ntac 12 (pop_assum kall_tac)
\\ rveq
\\ simp[mlmapTheory.empty_thm])))
QED
Theorem lookup_union_left_none:
!m1 m2.
map_ok m1 /\
map_ok m2 /\
cmp_of m1 = cmp_of m2 /\
mlmap$lookup m1 k = NONE ==>
mlmap$lookup m2 k = mlmap$lookup (mlmap$union m1 m2) k
Proof
cases_on `m1`
\\ cases_on `m2`
\\ strip_tac
\\ imp_res_tac mlmapTheory.union_thm
\\ imp_res_tac mlmapTheory.lookup_thm
\\ res_tac
\\ fs[mlmapTheory.lookup_thm]
\\ fs[FLOOKUP_FUNION]
QED
Theorem lookup_union_right_none:
!m1 m2.
map_ok m1 /\
map_ok m2 /\
cmp_of m1 = cmp_of m2 /\
mlmap$lookup m2 k = NONE ==>
mlmap$lookup m1 k = mlmap$lookup (mlmap$union m1 m2) k
Proof
cases_on `m2`
\\ cases_on `m1`
\\ strip_tac
\\ `map_ok (union (Map f' b') (Map f b))` by (imp_res_tac mlmapTheory.union_thm)
\\ `to_fmap (union (Map f' b') (Map f b)) =
to_fmap (Map f' b') ⊌ to_fmap (Map f b)`
by (imp_res_tac mlmapTheory.union_thm \\ fs[])
\\ fs[mlmapTheory.lookup_thm]
\\ fs[FLOOKUP_FUNION]
\\ CASE_TAC
\\ imp_res_tac mlmapTheory.lookup_thm
\\ res_tac
\\ fs[]
QED
Theorem lookup_list_union_none:
!xs.
EVERY (\m. lookup m k = NONE) xs /\ xs <> [] /\
EVERY map_ok xs /\
EVERY (\t. cmp_of t = cmp) xs ==>
lookup (list_union xs) k = NONE
Proof
Induct
>-(fs[])
>-(rpt strip_tac
\\imp_res_tac list_union_thm
\\simp[mlmapTheory.lookup_thm]
\\Cases_on `xs=[]`
>-(rfs[]
\\fs[mlmapTheory.lookup_thm])
>-(rfs[]
\\res_tac
\\Cases_on `xs`
>-(fs[])
>-( `map_ok (list_union (h'::t))` by (rw[] \\imp_res_tac list_union_thm
\\rfs[LENGTH_NIL])
\\imp_res_tac mlmapTheory.lookup_thm
\\simp[FLOOKUP_FUNION]
\\Cases_on `FLOOKUP (to_fmap h) k = NONE`
>-(fs[])
>-(fs[])
)))
QED
Theorem to_fmap_list_union_APPEND:
!xs ys. xs <> [] /\ ys <> [] /\
EVERY map_ok (xs++ys) /\
EVERY (\t. cmp_of t = cmp) (xs++ys) ==>
mlmap$to_fmap (list_union (xs++ys)) =
FUNION (mlmap$to_fmap (list_union xs)) (to_fmap (list_union ys))
Proof
Induct
>-(fs[])
\\rpt strip_tac
\\ cases_on `xs`
\\fs[]
>- (cases_on `ys`
>- (fs[])
>- (`map_ok (list_union (h'::t))` by fs[list_union_map_ok]
\\`cmp_of (list_union (h'::t)) = cmp` by fs[list_union_cmp_of]
\\fs[list_union_def, mlmapTheory.union_thm]))
>- (fs[list_union_def]
\\`map_ok (list_union (h'::t))` by fs[list_union_map_ok]
\\`cmp_of (list_union (h'::t)) = cmp` by fs[list_union_cmp_of]
\\`map_ok (list_union (h'::(t ++ ys)))` by fs[list_union_map_ok]
\\`cmp_of (list_union (h'::(t ++ ys))) = cmp` by fs[list_union_cmp_of]
\\fs[mlmapTheory.union_thm, FUNION_ASSOC])
QED
Theorem lookup_list_union_append_eq_right:
!xs ys cmp.
EVERY map_ok (xs++ys) /\
EVERY (\t. cmp_of t = cmp) (xs++ys) /\
EVERY (\m. lookup m k = NONE) xs /\ ys <> [] ==>
lookup (list_union (xs ++ ys)) k = lookup (list_union ys) k
Proof
Induct
\\ rpt strip_tac
>- (simp[])
\\ rpt strip_tac
\\ fs[list_union_def]
\\ cases_on `xs`
>- (res_tac
\\ fs[] \\ rw[]
\\ cases_on `ys`
>- ( fs[] )
>- ( fs[list_union_def]
\\ rw[]
\\ `EVERY (\t. cmp_of t = cmp_of h) (h'::t)`
by (imp_res_tac EVERY_DEF \\ fs[])
\\ `EVERY map_ok (h'::t)` by (imp_res_tac EVERY_DEF)
\\ `cmp_of (list_union (h'::t)) = cmp_of h`
by (imp_res_tac list_union_cmp_of \\ fs[])
\\ `map_ok (list_union (h'::t))` by (imp_res_tac list_union_map_ok \\ fs[])
\\ fs[lookup_union_left_none]))
>- (fs[list_union_def]
\\ rw[]
\\ cases_on `ys`
>- ( fs[])
>- (
`lookup (list_union (h'::(t ++ h''::t'))) k =
lookup (list_union (h''::t')) k` by res_tac
\\ rw[]
\\ `EVERY (\t. cmp_of t = cmp_of h) (h'::(t ++ h''::t'))`
by (imp_res_tac EVERY_DEF \\ fs[EVERY_APPEND])
\\ `EVERY map_ok (h'::(t ++ h''::t'))` by (imp_res_tac EVERY_DEF \\ fs[EVERY_APPEND])
\\ `cmp_of (list_union (h'::(t ++ h''::t'))) = cmp_of h`
by (imp_res_tac list_union_cmp_of \\ fs[])
\\ `map_ok (list_union (h'::(t ++ h''::t')))`
by (imp_res_tac list_union_map_ok \\ fs[])
\\ Cases_on `list_union (h'::(t ++ h''::t'))`
\\ imp_res_tac lookup_union_left_none
\\ rfs[]))
QED
Theorem lookup_list_union_append_eq_left:
!xs ys cmp.
EVERY map_ok (xs++ys) /\
EVERY (\t. cmp_of t = cmp) (xs++ys) /\
EVERY (\m. lookup m k = NONE) ys /\ xs <> [] ==>
lookup (list_union (xs ++ ys)) k = lookup (list_union xs) k
Proof
rpt strip_tac
\\ fs[]
\\ `EVERY (λt. cmp_of t = cmp) (xs ++ ys)` by fs[EVERY_APPEND]
\\ `EVERY map_ok (xs ++ ys)` by fs[EVERY_APPEND]
\\ `map_ok (list_union xs)`
by (cases_on `xs` >- (fs[]) >- (fs[list_union_map_ok]))
\\ `cmp_of (list_union xs) = cmp`
by (cases_on `xs` >- (fs[]) >- (fs[list_union_cmp_of]))
\\ `map_ok (list_union (xs++ys))`
by(cases_on `xs++ys` >- (fs[]) >- (imp_res_tac list_union_map_ok \\ fs[]))
\\ fs[mlmapTheory.lookup_thm] \\ rw[]
\\ cases_on `ys`
>- (fs[])
>- (
fs[to_fmap_list_union_APPEND]
\\ cases_on `FLOOKUP (to_fmap (list_union xs)) k`
>- (fs[FLOOKUP_FUNION]
\\ `map_ok (list_union (h::t))` by fs[list_union_map_ok]
\\ fs[GSYM mlmapTheory.lookup_thm]
\\ fs[lookup_list_union_none]
)
>- (fs[FLOOKUP_FUNION]))
QED
Theorem lookup_list_union_append_eq_mid:
!xs ys cmp.
EVERY map_ok (xs++[b]++ys) /\
EVERY (\t. cmp_of t = cmp) (xs++[b]++ys) /\
EVERY (\m. lookup m k = NONE) ys /\
EVERY (\m. lookup m k = NONE) xs ==>
lookup (list_union (xs ++ [b] ++ ys)) k = lookup b k
Proof
fs[lookup_list_union_append_eq_left, lookup_list_union_append_eq_right,
list_union_def]
QED
Theorem lookup_wrong_hash_none:
!(bs : ('a,'b) map list) i.
i <> (hf k) MOD (LENGTH bs) /\
buckets_ok bs hf /\
i < LENGTH bs ==>
(lookup (EL i bs) k = NONE)
Proof
rpt strip_tac
\\cases_on `lookup (EL i bs) k`
>- (fs[])
>- (fs[buckets_ok_def, hash_key_set_def, bucket_ok_def]
\\res_tac
\\fs[])
QED
Theorem lookup_left_right_none:
!bs.
0 < LENGTH bs /\
i = (hf k) MOD (LENGTH bs) /\
buckets_ok bs hf ==>
EVERY (\m. lookup m k = NONE) (TAKE i bs) /\
EVERY (\m. lookup m k = NONE) (DROP (i+1) bs)
Proof
strip_tac \\ strip_tac
\\`i < LENGTH bs` by fs[MOD_LESS]
\\`!j. j < (i :num) ==> (mlmap$lookup (EL j (TAKE i bs)) (k:'a)) = NONE`
by (simp[EL_TAKE]
\\ assume_tac lookup_wrong_hash_none
\\ res_tac
\\ fs[])
\\`!j. j < LENGTH bs − (i + 1) ==> (lookup (EL j (DROP (i+1) bs)) (k:'a)) = NONE`
by ( fs[EL_DROP] \\ assume_tac lookup_wrong_hash_none \\ fs[])
\\simp[EVERY_EL]
QED
Theorem lookup_list_union:
!bs k i cmp hf.
bs <> [] /\
i = hf k MOD LENGTH bs /\
buckets_ok bs hf /\
EVERY map_ok bs /\
EVERY (\b. cmp_of b = cmp) bs ==>
mlmap$lookup (list_union bs) k = mlmap$lookup (EL i bs) k
Proof
rpt strip_tac
\\ `0 < LENGTH bs` by fs[NOT_NIL_EQ_LENGTH_NOT_0]
\\ `i < LENGTH bs` by fs[MOD_LESS]
\\ `i+1 <= LENGTH bs` by fs[MOD_LESS]
\\ `?xs ys.
(xs ++ [EL i bs] ++ ys) = bs /\
EVERY map_ok (xs ++ [EL i bs] ++ ys) /\
EVERY (\m. cmp_of m = cmp) (xs++[EL i bs]++ys) /\
EVERY (\m. lookup m k = NONE) xs /\
EVERY (\m. lookup m k = NONE) ys`
by (qexists_tac `TAKE i bs`
\\qexists_tac `DROP (i+1) bs`
\\ conj_tac >- (imp_res_tac TAKE_SEG_DROP \\ rfs[SEG1])
\\ imp_res_tac lookup_left_right_none
\\ simp[]
\\ simp[EVERY_EL, EVERY_DEF, EVERY_TAKE, EVERY_DROP, EVERY_APPEND]
\\ rfs[EVERY_EL])
\\ `bs = xs ++ [EL i bs] ++ ys` by simp[]
\\ `lookup (list_union (xs++[EL i bs]++ys)) k = lookup (EL i bs) k`
by imp_res_tac lookup_list_union_append_eq_mid
\\ rfs[]
QED
Theorem replicate_every_same:
(REPLICATE n e) = xs ==> EVERY ($=e) xs
Proof
strip_tac
\\ rw[]
\\ fs[EVERY_REPLICATE]
QED
Theorem replicate_empty_map_thm:
!cmp.
TotOrd cmp /\
(REPLICATE n (mlmap$empty cmp)) = xs ==>
EVERY ($=(mlmap$empty cmp)) xs /\
EVERY (mlmap$map_ok) xs /\
EVERY (\t. mlmap$cmp_of t = cmp) xs
Proof
rpt strip_tac
>-(imp_res_tac replicate_every_same)
>-(imp_res_tac replicate_every_same
\\rw[]
\\fs[EVERY_REPLICATE, mlmapTheory.empty_thm])
>-(imp_res_tac replicate_every_same
\\rw[]
\\fs[EVERY_REPLICATE, mlmapTheory.cmp_of_def])
QED
Theorem hashtable_empty_spec:
!a b hf hfv cmp cmpv size sizev ar.
NUM size sizev /\
(a --> NUM) hf hfv /\
(a --> a --> ORDERING_TYPE) cmp cmpv /\
TotOrd cmp ==>
app (p:'ffi ffi_proj) Hashtable_empty_v [sizev; hfv; cmpv]
emp
(POSTv htv. HASHTABLE a b hf cmp FEMPTY htv)
Proof
rpt strip_tac
\\ xcf_with_def Hashtable_empty_v_def
\\ xlet_auto
>-(xsimpl)
\\ xlet `POSTv v. &(NUM 1 v \/ (NUM size' v /\ BOOL F bv))`
THEN1 (xif
\\ xlit
\\ xsimpl
\\ fs[BOOL_def])
(*size > 1*)
THEN1 (xlet `POSTv ar. SEP_EXISTS mpv. &(MAP_TYPE a b (mlmap$empty cmp) mpv) *
ARRAY ar (REPLICATE 1 mpv)`
>-(xapp \\ simp[])
\\ xlet `POSTv loc. SEP_EXISTS addr arr. &(addr = loc) * REF_ARRAY loc arr (REPLICATE 1 mpv)`
>-(xref
\\ fs[REF_ARRAY_def,REF_NUM_def]
\\ xsimpl)
\\ xlet `POSTv loc. SEP_EXISTS arr. REF_NUM loc 0 * REF_ARRAY addr arr (REPLICATE 1 mpv)`
>-(xref
\\ fs[REF_ARRAY_def, REF_NUM_def]
\\ xsimpl)
\\ xcon
\\ fs[HASHTABLE_def, hashtable_con_stamp_def]
\\ xsimpl
\\ qexists_tac `(REPLICATE 1 mpv)`
\\ qexists_tac `arr`
\\ qexists_tac `0`
\\ xsimpl
\\ fs[hashtable_inv_def]
\\ qexists_tac `(REPLICATE 1 (mlmap$empty cmp))`
\\ rpt conj_tac
THEN1(simp[REPLICATE_GENLIST, GENLIST_CONS, list_union_def, mlmapTheory.empty_thm])
THEN1(simp[buckets_ok_empty])
THEN1(simp[LIST_REL_REPLICATE_same])
THEN1(fs[EVERY_EL,HD, REPLICATE_GENLIST, GENLIST_CONS, mlmapTheory.empty_thm, balanced_mapTheory.empty_thm])
\\fs[EVERY_EL,HD, REPLICATE_GENLIST, GENLIST_CONS, mlmapTheory.cmp_of_def])
(*size > 1*)
\\ xlet `POSTv ar. SEP_EXISTS mpv. &(MAP_TYPE a b (mlmap$empty cmp) mpv) *
ARRAY ar (REPLICATE size' mpv)`
>-(xapp \\ simp[])
\\ xlet `POSTv loc. SEP_EXISTS addr arr. &(addr = loc) *
REF_ARRAY loc arr (REPLICATE size' mpv)`
>-(xref
\\fs[REF_ARRAY_def,REF_NUM_def]
\\ xsimpl)
\\ xlet `POSTv loc. SEP_EXISTS arr. REF_NUM loc 0 * REF_ARRAY addr arr (REPLICATE size' mpv)`
>-(xref
\\ fs[REF_ARRAY_def, REF_NUM_def]
\\ xsimpl)
\\ xcon
\\ fs[HASHTABLE_def, hashtable_con_stamp_def]
\\ xsimpl
\\ qexists_tac `(REPLICATE size' mpv)`
\\ qexists_tac `arr`
\\ qexists_tac `0`
\\ xsimpl
\\ fs[hashtable_inv_def]
\\ qexists_tac `(REPLICATE size' (mlmap$empty cmp))`
\\ rpt conj_tac
THEN1 (Cases_on `REPLICATE size' (mlmap$empty cmp)`
>-(fs[BOOL_def, REPLICATE_NIL])
\\imp_res_tac replicate_empty_map_thm
\\fs[list_union_empty_maps])
THEN1 (simp[buckets_ok_empty])
THEN1 (fs[BOOL_def, REPLICATE_NIL])
THEN1 (simp[LIST_REL_REPLICATE_same])
THEN1 (Cases_on `REPLICATE size' (mlmap$empty cmp)`
>-(fs[BOOL_def, REPLICATE_NIL])
\\ imp_res_tac replicate_empty_map_thm)
THEN1 (Cases_on `REPLICATE size' (mlmap$empty cmp)`
>-(fs[BOOL_def, REPLICATE_NIL])
\\imp_res_tac replicate_empty_map_thm)
QED
Theorem buckets_ok_insert:
!buckets hf idx k v.
EVERY map_ok buckets /\
buckets_ok buckets hf /\
idx = hf k MOD LENGTH buckets ==>
buckets_ok
(LUPDATE (mlmap$insert (EL idx buckets) k v)
idx buckets) hf
Proof
rpt strip_tac
\\fs[EVERY_EL,EL_LUPDATE,buckets_ok_def, bucket_ok_def, hash_key_set_def]
\\ntac 4 strip_tac
\\Cases_on `i = hf k MOD LENGTH buckets`
\\fs[mlmapTheory.lookup_insert]
\\Cases_on `k=k'`
\\ntac 3 (simp[])
QED
Theorem buckets_ok_delete:
!buckets hf idx k v.
EVERY map_ok buckets /\
buckets_ok buckets hf /\
idx = hf k MOD LENGTH buckets ==>
buckets_ok
(LUPDATE (mlmap$delete (EL idx buckets) k)
idx buckets) hf
Proof
rpt strip_tac
\\fs[EVERY_EL,EL_LUPDATE,buckets_ok_def, bucket_ok_def, hash_key_set_def]
\\ntac 4 strip_tac
\\Cases_on `i = hf k MOD LENGTH buckets`
\\fs[mlmapTheory.lookup_delete]
\\Cases_on `k=k'`
\\ntac 3 (simp[])
QED
Theorem insert_not_empty:
!a b (mp:('a,'b) map) k v.
mlmap$map_ok mp ==>
to_fmap (mlmap$insert mp k v) <> FEMPTY
Proof
fs[mlmapTheory.insert_thm, mlmapTheory.to_fmap_def,
balanced_mapTheory.insert_thm,
balanced_mapTheory.to_fmap_def, FEMPTY_FUPDATE_EQ]
QED
Theorem list_rel_insert:
LIST_REL (MAP_TYPE a b) buckets vlv /\
MAP_TYPE a b (mlmap$insert (EL idx buckets) k v) updMap /\
EVERY map_ok buckets /\
idx < LENGTH buckets ==>
LIST_REL (MAP_TYPE a b)
(LUPDATE (mlmap$insert (EL idx buckets) k v) idx buckets)
(LUPDATE updMap idx vlv)
Proof
rpt strip_tac
\\ fs[EVERY_EL,LIST_REL_EL_EQN, EL_LUPDATE]
\\ strip_tac
\\ strip_tac
\\ Cases_on `n = idx`
\\ fs[mlmapTheory.insert_thm]
\\ simp[]
QED
Theorem list_rel_delete:
LIST_REL (MAP_TYPE a b) buckets vlv /\
MAP_TYPE a b (mlmap$delete (EL idx buckets) k) updMap /\
EVERY map_ok buckets /\
idx < LENGTH buckets ==>
LIST_REL (MAP_TYPE a b)
(LUPDATE (mlmap$delete (EL idx buckets) k) idx buckets)
(LUPDATE updMap idx vlv)
Proof
rpt strip_tac
\\ fs[EVERY_EL,LIST_REL_EL_EQN, EL_LUPDATE]
\\ strip_tac
\\ strip_tac
\\ Cases_on `n = idx`
\\ fs[mlmapTheory.delete_thm]
\\ simp[]
QED
Theorem every_map_ok_insert:
!buckets idx k v.
EVERY map_ok buckets /\
idx < LENGTH buckets ==>
EVERY map_ok (LUPDATE (insert (EL idx buckets) k v) idx buckets)
Proof
rpt strip_tac
\\ fs[EVERY_EL,EL_LUPDATE]
\\ strip_tac
\\ strip_tac
\\ Cases_on `n=idx`
\\ fs[mlmapTheory.insert_thm]
\\ simp[]
QED
Theorem every_map_ok_delete:
!buckets idx k v.
EVERY map_ok buckets /\
idx < LENGTH buckets ==>
EVERY map_ok (LUPDATE (mlmap$delete (EL idx buckets) k) idx buckets)
Proof
rpt strip_tac
\\ fs[EVERY_EL,EL_LUPDATE]
\\ strip_tac
\\ strip_tac
\\ Cases_on `n=idx`
\\ fs[mlmapTheory.delete_thm]
\\ simp[]
QED
Theorem list_union_lupdate_insert:
bs <> [] /\
buckets_ok bs hf /\
EVERY map_ok bs /\
EVERY (\m. cmp_of m = cmp) bs ==>
to_fmap (list_union bs) |+ (k,v) =
to_fmap (list_union (LUPDATE (insert (EL (hf k MOD LENGTH bs) bs) k v)
(hf k MOD LENGTH bs) bs))
Proof
rpt strip_tac
\\ rfs[fmap_eq_flookup] \\ strip_tac
\\ `?i. i = hf k MOD LENGTH bs` by simp[]
\\ `?j. j = hf x MOD LENGTH bs` by simp[]
\\ `0 < LENGTH bs` by fs[NOT_NIL_EQ_LENGTH_NOT_0]
\\ `i < (LENGTH bs)` by fs[MOD_LESS]
\\ `map_ok (list_union bs)`
by (cases_on `bs` >- (fs[]) >- (imp_res_tac list_union_map_ok))
\\ `map_ok (EL i bs) /\ cmp_of (EL i bs) = cmp` by fs[EVERY_EL]
\\ `map_ok (insert (EL i bs) k v)` by fs[mlmapTheory.insert_thm]
\\ `cmp_of (insert (EL i bs) k v) = cmp` by fs[mlmapTheory.cmp_of_insert]
\\ `EVERY map_ok (LUPDATE (insert (EL i bs) k v) i bs) /\
EVERY (\m. cmp_of m = cmp) (LUPDATE (insert (EL i bs) k v) i bs)`
by fs[IMP_EVERY_LUPDATE]
\\ `map_ok (list_union(LUPDATE (insert (EL i bs) k v) i bs))`
by (cases_on `LUPDATE (insert (EL i bs) k v) i bs`
>- (fs[])
>- (imp_res_tac list_union_map_ok \\ fs[]))
\\ `buckets_ok (LUPDATE (insert (EL i bs) k v) i bs) hf` by fs[buckets_ok_insert]
\\ `lookup (list_union (LUPDATE (insert (EL i bs) k v) i bs)) x
= lookup (EL j (LUPDATE (insert (EL i bs) k v) i (bs : ('a,'b) map list))) x`
by (`(LUPDATE (insert (EL i bs) k v) i bs) <> []`
by fs[LENGTH_LUPDATE, NOT_NIL_EQ_LENGTH_NOT_0]
\\ `j = hf (x:'a) MOD (LENGTH (LUPDATE (insert (EL i bs) k v) i bs))`
by fs[LENGTH_LUPDATE]
\\ imp_res_tac lookup_list_union)
\\ rfs[GSYM mlmapTheory.lookup_thm]
\\ cases_on `x=k`
>- (fs[EL_LUPDATE, mlmapTheory.lookup_insert, FLOOKUP_UPDATE])
>- (fs[FLOOKUP_UPDATE]
\\simp[GSYM mlmapTheory.lookup_thm,EL_LUPDATE]
\\CASE_TAC
>-(simp[mlmapTheory.lookup_insert]
\\`?j'. j' = hf x MOD LENGTH bs` by simp[]
\\imp_res_tac lookup_list_union
\\simp[])
>-(imp_res_tac lookup_list_union \\ simp[]))
QED
Theorem list_union_lupdate_delete:
bs <> [] /\
buckets_ok bs hf /\
EVERY map_ok bs /\
EVERY (\m. cmp_of m = cmp) bs ==>
to_fmap (list_union bs) \\ k =
to_fmap (list_union (LUPDATE (mlmap$delete
(EL (hf k MOD LENGTH bs) bs) k)
(hf k MOD LENGTH bs) bs))
Proof
rpt strip_tac
\\ rfs[fmap_eq_flookup] \\ strip_tac
\\ `?i. i = hf k MOD LENGTH bs` by simp[]
\\ `?j. j = hf x MOD LENGTH bs` by simp[]
\\ `0 < LENGTH bs` by fs[NOT_NIL_EQ_LENGTH_NOT_0]
\\ `i < (LENGTH bs)` by fs[MOD_LESS]
\\ `map_ok (list_union bs)`
by (cases_on `bs` >- (fs[]) >- (imp_res_tac list_union_map_ok))
\\ `map_ok (EL i bs) /\ cmp_of (EL i bs) = cmp` by fs[EVERY_EL]
\\ `map_ok (delete (EL i bs) k)` by fs[mlmapTheory.delete_thm]
\\ `cmp_of (delete (EL i bs) k) = cmp` by fs[mlmapTheory.cmp_of_delete]
\\ `EVERY map_ok (LUPDATE (delete (EL i bs) k) i bs) /\
EVERY (\m. cmp_of m = cmp) (LUPDATE (delete (EL i bs) k) i bs)`
by fs[IMP_EVERY_LUPDATE]
\\ `map_ok (list_union(LUPDATE (delete (EL i bs) k) i bs))`
by (cases_on `LUPDATE (delete (EL i bs) k) i bs`
>- (fs[])
>- (imp_res_tac list_union_map_ok \\ fs[]))
\\ `buckets_ok (LUPDATE (delete (EL i bs) k) i bs) hf` by fs[buckets_ok_delete]
\\ `lookup (list_union (LUPDATE (delete (EL i bs) k) i bs)) x
= lookup (EL j (LUPDATE (delete (EL i bs) k) i (bs : ('a,'b) map list))) x`
by (`(LUPDATE (delete (EL i bs) k) i bs) <> []`
by fs[LENGTH_LUPDATE, NOT_NIL_EQ_LENGTH_NOT_0]
\\ `j = hf (x:'a) MOD (LENGTH (LUPDATE (delete (EL i bs) k) i bs))`
by fs[LENGTH_LUPDATE]
\\ imp_res_tac lookup_list_union)
\\ rfs[GSYM mlmapTheory.lookup_thm]
\\ cases_on `x=k`
>- (fs[EL_LUPDATE, mlmapTheory.lookup_delete, DOMSUB_FLOOKUP_THM])
>- (fs[DOMSUB_FLOOKUP_THM]
\\simp[GSYM mlmapTheory.lookup_thm,EL_LUPDATE]
\\CASE_TAC
>-(simp[mlmapTheory.lookup_delete]
\\`?j'. j' = hf x MOD LENGTH bs` by simp[]
\\imp_res_tac lookup_list_union
\\simp[])
>-(imp_res_tac lookup_list_union \\ simp[]))
QED
Theorem every_cmp_of_insert:
!buckets idx k v.
EVERY (λt. cmp_of t = cmp) buckets /\
idx < LENGTH buckets ==>
EVERY (λt. cmp_of t = cmp) (LUPDATE (insert (EL idx buckets) k v) idx buckets)
Proof
rpt strip_tac
\\ fs[EVERY_EL,EL_LUPDATE]
\\ strip_tac
\\ strip_tac
\\ Cases_on `n=idx`
\\ fs[mlmapTheory.insert_thm]
\\ simp[]
QED
Theorem every_cmp_of_delete:
!buckets idx k v.
EVERY (λt. cmp_of t = cmp) buckets /\
idx < LENGTH buckets ==>
EVERY (λt. cmp_of t = cmp)
(LUPDATE (mlmap$delete
(EL idx buckets) k) idx buckets)
Proof
rpt strip_tac
\\ fs[EVERY_EL,EL_LUPDATE]
\\ strip_tac
\\ strip_tac
\\ Cases_on `n=idx`
\\ fs[mlmapTheory.delete_thm]
\\ simp[]
QED
Theorem hashtable_staticInsert_spec:
!a b hf cmp k kv v vv htv.
a k kv /\
b v vv ==>
app (p:'ffi ffi_proj) Hashtable_staticInsert_v [htv; kv; vv]
(HASHTABLE a b hf cmp h htv)
(POSTv uv. &(UNIT_TYPE () uv) * HASHTABLE a b hf cmp (h|+(k,v)) htv)
Proof
rpt strip_tac
\\ xcf_with_def Hashtable_staticInsert_v_def
\\ fs[HASHTABLE_def, hashtable_con_stamp_def]
\\ xpull
\\ xmatch
\\ fs[hashtable_inv_def]
\\ xlet `POSTv arr. SEP_EXISTS aRef arr2 avl uRef uv uvv.
&(aRef = ar /\ arr2 = arr /\ avl = vlv /\ uRef = ur /\ uv = heuristic_size) *
REF_ARRAY ar arr vlv * REF ur uvv * & (NUM heuristic_size uvv)`
>-(xapp
\\qexists_tac `ARRAY arr vlv * REF_NUM ur heuristic_size`
\\qexists_tac `arr`
\\fs[REF_ARRAY_def, REF_NUM_def]
\\xsimpl)
\\ xlet `POSTv v. SEP_EXISTS length. &(length = LENGTH vlv /\ NUM length v) * REF_ARRAY aRef arr2 avl * REF_NUM uRef uv`
>-(xapp
\\qexists_tac `aRef ~~> arr2 * REF_NUM uRef uv`
\\qexists_tac `avl`
\\fs[REF_ARRAY_def,REF_NUM_def]
\\xsimpl)
\\ xlet `POSTv v. SEP_EXISTS hashval. &(hashval = (hf k) /\ NUM hashval v) * REF_ARRAY aRef arr2 avl * REF_NUM uRef uv`
>-(xapp
\\qexists_tac `REF_ARRAY aRef arr2 avl * REF_NUM uRef uv`
\\qexists_tac `k`
\\qexists_tac `hf`
\\conj_tac
>-(qexists_tac `a`
\\simp[])
\\xsimpl)
\\ xlet `POSTv v. SEP_EXISTS idx. &(idx = (hashval MOD length') /\ NUM idx v /\
idx < LENGTH avl /\ idx < LENGTH buckets /\ LENGTH buckets = LENGTH avl) * REF_ARRAY aRef arr2 avl * REF_NUM uRef uv`
>-(xapp
\\qexists_tac `REF_ARRAY aRef arr2 avl * REF_NUM uRef uv`
\\qexists_tac `&length'`
\\qexists_tac `&hashval`
\\fs[NOT_NIL_EQ_LENGTH_NOT_0,X_MOD_Y_EQ_X,LENGTH_NIL_SYM,NUM_def, hashtable_inv_def, LIST_REL_LENGTH]
\\xsimpl
\\EVAL_TAC
\\fs[LIST_REL_LENGTH,NOT_NIL_EQ_LENGTH_NOT_0,LIST_REL_EL_EQN, X_MOD_Y_EQ_X])
\\ xlet `POSTv mp. SEP_EXISTS oldMap. &(oldMap = mp /\ MAP_TYPE a b (EL idx buckets) mp) * REF_ARRAY aRef arr2 avl * REF_NUM uRef uv`
>-(xapp
\\qexists_tac `aRef ~~> arr2 * REF_NUM uRef uv`
\\qexists_tac `idx`
\\qexists_tac `vlv`
\\fs[hashtable_inv_def,NOT_NIL_EQ_LENGTH_NOT_0,LIST_REL_EL_EQN, X_MOD_Y_EQ_X,REF_ARRAY_def]
\\xsimpl)
\\ xlet `POSTv retv. SEP_EXISTS newMp.
&(newMp = retv /\ MAP_TYPE a b (mlmap$insert (EL idx buckets) k v) retv) * REF_ARRAY aRef arr2 avl * REF_NUM uRef uv`
>-(xapp
\\qexists_tac `REF_ARRAY aRef arr2 avl * REF_NUM uRef uv`
\\qexists_tac `v`
\\qexists_tac `k`
\\qexists_tac `EL idx buckets`
\\qexists_tac `b`
\\qexists_tac `a`
\\fs[LIST_REL_EL_EQN]
\\xsimpl)
\\ xlet `POSTv unitv. SEP_EXISTS newBuckets.
&(UNIT_TYPE () unitv /\ newBuckets = (LUPDATE newMp idx avl)) *
REF_ARRAY aRef arr2 newBuckets * REF_NUM uRef uv`
>-(xapp
\\qexists_tac `aRef ~~> arr2 * REF_NUM uRef uv`
\\qexists_tac `idx`
\\qexists_tac `vlv`
\\fs[REF_ARRAY_def]
\\xsimpl)
\\ xlet `POSTv b. &(BOOL (mlmap$null (EL idx buckets)) b) * REF_ARRAY aRef arr2 newBuckets * REF_NUM uRef uv`
>-(xapp
\\qexists_tac `REF_ARRAY aRef arr2 newBuckets * REF_NUM uRef uv`
\\qexists_tac `EL idx buckets`
\\xsimpl
\\qexists_tac `a`
\\qexists_tac `b`
\\fs[])
THEN1 (xif
THEN1 (xlet `POSTv usedv. &(NUM uv usedv) * REF_ARRAY aRef arr2 newBuckets * REF_NUM uRef uv`
>-(xapp
\\qexists_tac `REF_ARRAY aRef arr2 newBuckets`
\\qexists_tac `uvv`
\\fs[REF_NUM_def, NUM_def, INT_def]
\\xsimpl)
\\ xlet_auto
>-(qexists_tac `REF_ARRAY aRef arr2 newBuckets * REF_NUM uRef uv`
\\xsimpl)
THEN1( xapp
\\qexists_tac `REF_ARRAY aRef arr2 newBuckets`
\\qexists_tac `usedv`
\\fs[REF_NUM_def, NUM_def, INT_def]
\\xsimpl
\\strip_tac
\\strip_tac
\\qexists_tac `uRef`
\\qexists_tac `aRef`
\\qexists_tac `hfv`
\\qexists_tac `newBuckets`
\\qexists_tac `arr2`
\\qexists_tac `cmpv`
\\qexists_tac `heuristic_size + 1`
\\xsimpl
\\qexists_tac `LUPDATE (mlmap$insert (EL (hf k MOD LENGTH buckets) buckets) k v) (hf k MOD LENGTH buckets) buckets`
\\`buckets <> []` by simp[NOT_NIL_EQ_LENGTH_NOT_0]
\\ imp_res_tac list_union_lupdate_insert
\\ simp[]
\\fs[every_cmp_of_insert, buckets_ok_insert, list_rel_insert, every_map_ok_insert]))
\\xcon
\\xsimpl
\\qexists_tac `uRef`
\\qexists_tac `aRef`
\\qexists_tac `hfv`
\\qexists_tac `newBuckets`
\\qexists_tac `arr2`
\\qexists_tac `cmpv`
\\qexists_tac `heuristic_size`
\\xsimpl
\\qexists_tac `LUPDATE (mlmap$insert (EL (hf k MOD LENGTH buckets) buckets) k v) (hf k MOD LENGTH buckets) buckets`
\\`buckets <> []` by simp[NOT_NIL_EQ_LENGTH_NOT_0]
\\ imp_res_tac list_union_lupdate_insert
\\ simp[]
\\fs[every_cmp_of_insert, buckets_ok_insert, list_rel_insert, every_map_ok_insert])
QED
Theorem list_app_pairs_spec:
∀a b l start lv.
LIST_TYPE (PAIR_TYPE a b) l lv /\
(!n xv. n < LENGTH l ∧ (PAIR_TYPE a b) (EL n l) xv ==>
app p fv [xv] (eff (start+n)) (POSTv v. &UNIT_TYPE () v * (eff (start+n+1))))
==>
app (p:'ffi ffi_proj) List_app_v [fv; lv] (eff start)
(POSTv v. &UNIT_TYPE () v * (eff (start + (LENGTH l))))
Proof
Induct_on `l` \\ rw[LIST_TYPE_def] \\ rpt strip_tac
>- ( xcf_with_def List_app_v_def \\ xmatch \\ xcon \\ xsimpl )
\\ xcf_with_def List_app_v_def
\\ xmatch
\\ xlet`POSTv v. &UNIT_TYPE () v * eff (start+1)`
>- (
xapp
\\ CONV_TAC(RESORT_EXISTS_CONV(sort_vars["n"]))
\\ qexists_tac`0` \\ xsimpl )
\\ first_x_assum drule
\\ disch_then(qspec_then`start+1`mp_tac)
\\ simp[ADD1]
\\ impl_tac
>- (
rw[]
\\ first_x_assum(qspec_then`n+1`mp_tac)
\\ simp[EL_CONS,PRE_SUB1] )
\\ strip_tac \\ xapp
QED
Theorem fupdate_list_take_el:
n < LENGTH l ==>
fm|++ ((TAKE (n+1) l)) =
(fm|++ (TAKE n l)) |+ EL n l
Proof
strip_tac
\\fs[TAKE_EL_SNOC]
\\fs[SNOC_APPEND]
\\fs[FUPDATE_LIST_APPEND]
\\fs[FUPDATE_LIST_THM]
QED
Theorem hashtable_insertList_spec:
!a b hf cmp htv l lv.
LIST_TYPE (PAIR_TYPE a b) l lv ==>
app (p:'ffi ffi_proj) Hashtable_insertList_v [htv;lv]
(HASHTABLE a b hf cmp h htv)
(POSTv uv. &(UNIT_TYPE () uv) *
HASHTABLE a b hf cmp (h|++l) htv)
Proof
rpt strip_tac
\\ xcf_with_def Hashtable_insertList_v_def
\\xfun_spec `f`
`!k v pv h' n.
n < LENGTH l /\
(PAIR_TYPE a b) (EL n l) pv ==>
app (p:'ffi ffi_proj) f [pv]
(HASHTABLE a b hf cmp (h'|++TAKE n l) htv)
(POSTv uv. &(UNIT_TYPE () uv) *
HASHTABLE a b hf cmp ((h'|++TAKE n l)|+EL n l) htv)`