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|
signature testHashmapTheory =
sig
type thm = Thm.thm
(* Axioms *)
val insert_def : thm
(* Definitions *)
val distinct_keys_def : thm
val list_t_TY_DEF : thm
val list_t_case_def : thm
val list_t_size_def : thm
val list_t_v_def : thm
val lookup_def : thm
(* Theorems *)
val datatype_list_t : thm
val distinct_keys_cons : thm
val distinct_keys_insert : thm
val distinct_keys_insert_index_neq : thm
val distinct_keys_tail : thm
val insert_index_neq : thm
val insert_lem : thm
val insert_lem_aux : thm
val list_t_11 : thm
val list_t_Axiom : thm
val list_t_case_cong : thm
val list_t_case_eq : thm
val list_t_distinct : thm
val list_t_induction : thm
val list_t_nchotomy : thm
val lookup_raw_def : thm
val lookup_raw_ind : thm
val nth_mut_fwd_def : thm
val nth_mut_fwd_ind : thm
val nth_mut_fwd_spec : thm
val testHashmap_grammars : type_grammar.grammar * term_grammar.grammar
(*
[primitives] Parent theory of "testHashmap"
[insert_def] Axiom
[oracles: ] [axioms: insert_def] []
⊢ insert key value ls =
case ls of
ListCons (ckey,cvalue) tl =>
if ckey = key then Return (ListCons (ckey,value) tl)
else
do
tl0 <- insert key value tl;
Return (ListCons (ckey,cvalue) tl0)
od
| ListNil => Return (ListCons (key,value) ListNil)
[distinct_keys_def] Definition
⊢ ∀ls.
distinct_keys ls ⇔
∀i j.
0 ≤ i ⇒
i < j ⇒
j < len ls ⇒
FST (index i ls) ≠ FST (index j ls)
[list_t_TY_DEF] Definition
⊢ ∃rep.
TYPE_DEFINITION
(λa0'.
∀ $var$('list_t').
(∀a0'.
(∃a0 a1.
a0' =
(λa0 a1.
ind_type$CONSTR 0 a0
(ind_type$FCONS a1 (λn. ind_type$BOTTOM)))
a0 a1 ∧ $var$('list_t') a1) ∨
a0' =
ind_type$CONSTR (SUC 0) ARB (λn. ind_type$BOTTOM) ⇒
$var$('list_t') a0') ⇒
$var$('list_t') a0') rep
[list_t_case_def] Definition
⊢ (∀a0 a1 f v. list_t_CASE (ListCons a0 a1) f v = f a0 a1) ∧
∀f v. list_t_CASE ListNil f v = v
[list_t_size_def] Definition
⊢ (∀f a0 a1.
list_t_size f (ListCons a0 a1) = 1 + (f a0 + list_t_size f a1)) ∧
∀f. list_t_size f ListNil = 0
[list_t_v_def] Definition
⊢ list_t_v ListNil = [] ∧
∀x tl. list_t_v (ListCons x tl) = x::list_t_v tl
[lookup_def] Definition
⊢ ∀key ls. lookup key ls = lookup_raw key (list_t_v ls)
[datatype_list_t] Theorem
⊢ DATATYPE (list_t ListCons ListNil)
[distinct_keys_cons] Theorem
⊢ ∀k v ls.
(∀i. 0 ≤ i ⇒ i < len ls ⇒ FST (index i ls) ≠ k) ⇒
distinct_keys ls ⇒
distinct_keys ((k,v)::ls)
[distinct_keys_insert] Theorem
[oracles: DISK_THM] [axioms: insert_def] []
⊢ ∀k v ls0 ls1.
distinct_keys (list_t_v ls0) ⇒
insert k v ls0 = Return ls1 ⇒
distinct_keys (list_t_v ls1)
[distinct_keys_insert_index_neq] Theorem
[oracles: DISK_THM] [axioms: insert_def] []
⊢ ∀k v q r ls0 ls1 i.
distinct_keys ((q,r)::list_t_v ls0) ⇒
q ≠ k ⇒
insert k v ls0 = Return ls1 ⇒
0 ≤ i ⇒
i < len (list_t_v ls1) ⇒
FST (index i (list_t_v ls1)) ≠ q
[distinct_keys_tail] Theorem
⊢ ∀k v ls. distinct_keys ((k,v)::ls) ⇒ distinct_keys ls
[insert_index_neq] Theorem
[oracles: DISK_THM] [axioms: insert_def] []
⊢ ∀q k v ls0 ls1 i.
(∀j. 0 ≤ j ∧ j < len (list_t_v ls0) ⇒
q ≠ FST (index j (list_t_v ls0))) ⇒
q ≠ k ⇒
insert k v ls0 = Return ls1 ⇒
0 ≤ i ⇒
i < len (list_t_v ls1) ⇒
FST (index i (list_t_v ls1)) ≠ q
[insert_lem] Theorem
[oracles: DISK_THM] [axioms: insert_def] []
⊢ ∀ls key value.
distinct_keys (list_t_v ls) ⇒
case insert key value ls of
Return ls1 =>
lookup key ls1 = SOME value ∧
(∀k. k ≠ key ⇒ lookup k ls = lookup k ls1) ∧
distinct_keys (list_t_v ls1)
| Fail v1 => F
| Diverge => F
[insert_lem_aux] Theorem
[oracles: DISK_THM] [axioms: insert_def] []
⊢ ∀ls key value.
case insert key value ls of
Return ls1 =>
lookup key ls1 = SOME value ∧
∀k. k ≠ key ⇒ lookup k ls = lookup k ls1
| Fail v1 => F
| Diverge => F
[list_t_11] Theorem
⊢ ∀a0 a1 a0' a1'.
ListCons a0 a1 = ListCons a0' a1' ⇔ a0 = a0' ∧ a1 = a1'
[list_t_Axiom] Theorem
⊢ ∀f0 f1. ∃fn.
(∀a0 a1. fn (ListCons a0 a1) = f0 a0 a1 (fn a1)) ∧
fn ListNil = f1
[list_t_case_cong] Theorem
⊢ ∀M M' f v.
M = M' ∧ (∀a0 a1. M' = ListCons a0 a1 ⇒ f a0 a1 = f' a0 a1) ∧
(M' = ListNil ⇒ v = v') ⇒
list_t_CASE M f v = list_t_CASE M' f' v'
[list_t_case_eq] Theorem
⊢ list_t_CASE x f v = v' ⇔
(∃t l. x = ListCons t l ∧ f t l = v') ∨ x = ListNil ∧ v = v'
[list_t_distinct] Theorem
⊢ ∀a1 a0. ListCons a0 a1 ≠ ListNil
[list_t_induction] Theorem
⊢ ∀P. (∀l. P l ⇒ ∀t. P (ListCons t l)) ∧ P ListNil ⇒ ∀l. P l
[list_t_nchotomy] Theorem
⊢ ∀ll. (∃t l. ll = ListCons t l) ∨ ll = ListNil
[lookup_raw_def] Theorem
⊢ (∀key. lookup_raw key [] = NONE) ∧
∀v ls key k.
lookup_raw key ((k,v)::ls) =
if k = key then SOME v else lookup_raw key ls
[lookup_raw_ind] Theorem
⊢ ∀P. (∀key. P key []) ∧
(∀key k v ls. (k ≠ key ⇒ P key ls) ⇒ P key ((k,v)::ls)) ⇒
∀v v1. P v v1
[nth_mut_fwd_def] Theorem
⊢ ∀ls i.
nth_mut_fwd ls i =
case ls of
ListCons x tl =>
if u32_to_int i = 0 then Return x
else do i0 <- u32_sub i (int_to_u32 1); nth_mut_fwd tl i0 od
| ListNil => Fail Failure
[nth_mut_fwd_ind] Theorem
⊢ ∀P. (∀ls i.
(∀x tl i0. ls = ListCons x tl ∧ u32_to_int i ≠ 0 ⇒ P tl i0) ⇒
P ls i) ⇒
∀v v1. P v v1
[nth_mut_fwd_spec] Theorem
⊢ ∀ls i.
u32_to_int i < len (list_t_v ls) ⇒
case nth_mut_fwd ls i of
Return x => x = index (u32_to_int i) (list_t_v ls)
| Fail v1 => F
| Diverge => F
*)
end
|
