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diff --git a/tests/fstar-split/hashmap/Hashmap.Properties.fst b/tests/fstar-split/hashmap/Hashmap.Properties.fst new file mode 100644 index 00000000..def520f0 --- /dev/null +++ b/tests/fstar-split/hashmap/Hashmap.Properties.fst @@ -0,0 +1,3186 @@ +(** Properties about the hashmap *) +module Hashmap.Properties +open Primitives +open FStar.List.Tot +open FStar.Mul +open Hashmap.Types +open Hashmap.Clauses +open Hashmap.Funs + +#set-options "--z3rlimit 50 --fuel 0 --ifuel 1" + +let _align_fsti = () + +/// The proofs: +/// =========== +/// +/// The proof strategy is to do exactly as with Low* proofs (we initially tried to +/// prove more properties in one go, but it was a mistake): +/// - prove that, under some preconditions, the low-level functions translated +/// from Rust refine some higher-level functions +/// - do functional proofs about those high-level functions to prove interesting +/// properties about the hash map operations, and invariant preservation +/// - combine everything +/// +/// The fact that we work in a pure setting allows us to be more modular than when +/// working with effects. For instance we can do a case disjunction (see the proofs +/// for insert, which study the cases where the key is already/not in the hash map +/// in separate proofs - we had initially tried to do them in one step: it is doable +/// but requires some work, and the F* response time quickly becomes annoying while +/// making progress, so we split them). We can also easily prove a refinement lemma, +/// study the model, *then* combine those to also prove that the low-level function +/// preserves the invariants, rather than do everything at once as is usually the +/// case when doing intrinsic proofs with effects (I remember that having to prove +/// invariants in one go *and* a refinement step, even small, can be extremely +/// difficult in Low*). + + +(*** Utilities *) + +/// We need many small helpers and lemmas, mostly about lists (and the ones we list +/// here are not in the standard F* library). + +val index_append_lem (#a : Type0) (ls0 ls1 : list a) (i : nat{i < length ls0 + length ls1}) : + Lemma ( + (i < length ls0 ==> index (ls0 @ ls1) i == index ls0 i) /\ + (i >= length ls0 ==> index (ls0 @ ls1) i == index ls1 (i - length ls0))) + [SMTPat (index (ls0 @ ls1) i)] + +#push-options "--fuel 1" +let rec index_append_lem #a ls0 ls1 i = + match ls0 with + | [] -> () + | x :: ls0' -> + if i = 0 then () + else index_append_lem ls0' ls1 (i-1) +#pop-options + +val index_map_lem (#a #b: Type0) (f : a -> Tot b) (ls : list a) + (i : nat{i < length ls}) : + Lemma ( + index (map f ls) i == f (index ls i)) + [SMTPat (index (map f ls) i)] + +#push-options "--fuel 1" +let rec index_map_lem #a #b f ls i = + match ls with + | [] -> () + | x :: ls' -> + if i = 0 then () + else index_map_lem f ls' (i-1) +#pop-options + +val for_all_append (#a : Type0) (f : a -> Tot bool) (ls0 ls1 : list a) : + Lemma (for_all f (ls0 @ ls1) = (for_all f ls0 && for_all f ls1)) + +#push-options "--fuel 1" +let rec for_all_append #a f ls0 ls1 = + match ls0 with + | [] -> () + | x :: ls0' -> + for_all_append f ls0' ls1 +#pop-options + +/// Filter a list, stopping after we removed one element +val filter_one (#a : Type) (f : a -> bool) (ls : list a) : list a + +let rec filter_one #a f ls = + match ls with + | [] -> [] + | x :: ls' -> if f x then x :: filter_one f ls' else ls' + +val find_append (#a : Type) (f : a -> bool) (ls0 ls1 : list a) : + Lemma ( + find f (ls0 @ ls1) == + begin match find f ls0 with + | Some x -> Some x + | None -> find f ls1 + end) + +#push-options "--fuel 1" +let rec find_append #a f ls0 ls1 = + match ls0 with + | [] -> () + | x :: ls0' -> + if f x then + begin + assert(ls0 @ ls1 == x :: (ls0' @ ls1)); + assert(find f (ls0 @ ls1) == find f (x :: (ls0' @ ls1))); + // Why do I have to do this?! Is it because of subtyping?? + assert( + match find f (ls0 @ ls1) with + | Some x' -> x' == x + | None -> False) + end + else find_append f ls0' ls1 +#pop-options + +val length_flatten_update : + #a:Type + -> ls:list (list a) + -> i:nat{i < length ls} + -> x:list a -> + Lemma ( + // We want this property: + // ``` + // length (flatten (list_update ls i x)) = + // length (flatten ls) - length (index ls i) + length x + // ``` + length (flatten (list_update ls i x)) + length (index ls i) = + length (flatten ls) + length x) + +#push-options "--fuel 1" +let rec length_flatten_update #a ls i x = + match ls with + | x' :: ls' -> + assert(flatten ls == x' @ flatten ls'); // Triggers patterns + assert(length (flatten ls) == length x' + length (flatten ls')); + if i = 0 then + begin + let ls1 = x :: ls' in + assert(list_update ls i x == ls1); + assert(flatten ls1 == x @ flatten ls'); // Triggers patterns + assert(length (flatten ls1) == length x + length (flatten ls')); + () + end + else + begin + length_flatten_update ls' (i-1) x; + let ls1 = x' :: list_update ls' (i-1) x in + assert(flatten ls1 == x' @ flatten (list_update ls' (i-1) x)) // Triggers patterns + end +#pop-options + +val length_flatten_index : + #a:Type + -> ls:list (list a) + -> i:nat{i < length ls} -> + Lemma ( + length (flatten ls) >= length (index ls i)) + +#push-options "--fuel 1" +let rec length_flatten_index #a ls i = + match ls with + | x' :: ls' -> + assert(flatten ls == x' @ flatten ls'); // Triggers patterns + assert(length (flatten ls) == length x' + length (flatten ls')); + if i = 0 then () + else length_flatten_index ls' (i-1) +#pop-options + +val forall_index_equiv_list_for_all + (#a : Type) (pred : a -> Tot bool) (ls : list a) : + Lemma ((forall (i:nat{i < length ls}). pred (index ls i)) <==> for_all pred ls) + +#push-options "--fuel 1" +let rec forall_index_equiv_list_for_all pred ls = + match ls with + | [] -> () + | x :: ls' -> + assert(forall (i:nat{i < length ls'}). index ls' i == index ls (i+1)); + assert(forall (i:nat{0 < i /\ i < length ls}). index ls i == index ls' (i-1)); + assert(index ls 0 == x); + forall_index_equiv_list_for_all pred ls' +#pop-options + +val find_update: + #a:Type + -> f:(a -> Tot bool) + -> ls:list a + -> x:a + -> ls':list a{length ls' == length ls} +#push-options "--fuel 1" +let rec find_update #a f ls x = + match ls with + | [] -> [] + | hd::tl -> + if f hd then x :: tl else hd :: find_update f tl x +#pop-options + +val pairwise_distinct : #a:eqtype -> ls:list a -> Tot bool +let rec pairwise_distinct (#a : eqtype) (ls : list a) : Tot bool = + match ls with + | [] -> true + | x :: ls' -> List.Tot.for_all (fun y -> x <> y) ls' && pairwise_distinct ls' + +val pairwise_rel : #a:Type -> pred:(a -> a -> Tot bool) -> ls:list a -> Tot bool +let rec pairwise_rel #a pred ls = + match ls with + | [] -> true + | x :: ls' -> + for_all (pred x) ls' && pairwise_rel pred ls' + +#push-options "--fuel 1" +let rec flatten_append (#a : Type) (l1 l2: list (list a)) : + Lemma (flatten (l1 @ l2) == flatten l1 @ flatten l2) = + match l1 with + | [] -> () + | x :: l1' -> + flatten_append l1' l2; + append_assoc x (flatten l1') (flatten l2) +#pop-options + +/// We don't use anonymous functions as parameters to other functions, but rather +/// introduce auxiliary functions instead: otherwise we can't reason (because +/// F*'s encoding to the SMT is imprecise for functions) +let fst_is_disctinct (#a : eqtype) (#b : Type0) (p0 : a & b) (p1 : a & b) : Type0 = + fst p0 <> fst p1 + +(*** Lemmas about Primitives *) +/// TODO: move those lemmas + +// TODO: rename to 'insert'? +val list_update_index_dif_lem + (#a : Type0) (ls : list a) (i : nat{i < length ls}) (x : a) + (j : nat{j < length ls}) : + Lemma (requires (j <> i)) + (ensures (index (list_update ls i x) j == index ls j)) + [SMTPat (index (list_update ls i x) j)] + +#push-options "--fuel 1" +let rec list_update_index_dif_lem #a ls i x j = + match ls with + | x' :: ls -> + if i = 0 then () + else if j = 0 then () + else + list_update_index_dif_lem ls (i-1) x (j-1) +#pop-options + +val map_list_update_lem + (#a #b: Type0) (f : a -> Tot b) + (ls : list a) (i : nat{i < length ls}) (x : a) : + Lemma (list_update (map f ls) i (f x) == map f (list_update ls i x)) + [SMTPat (list_update (map f ls) i (f x))] + +#push-options "--fuel 1" +let rec map_list_update_lem #a #b f ls i x = + match ls with + | x' :: ls' -> + if i = 0 then () + else map_list_update_lem f ls' (i-1) x +#pop-options + +(*** Invariants, models *) + +(**** Internals *) +/// The following invariants, models, representation functions... are mostly +/// for the purpose of the proofs. + +let is_pos_usize (n : nat) : Type0 = 0 < n /\ n <= usize_max +type pos_usize = x:usize{x > 0} + +type binding (t : Type0) = key & t + +type slots_t (t : Type0) = alloc_vec_Vec (list_t t) + +/// We represent hash maps as associative lists +type assoc_list (t : Type0) = list (binding t) + +/// Representation function for [list_t] +let rec list_t_v (#t : Type0) (ls : list_t t) : assoc_list t = + match ls with + | List_Nil -> [] + | List_Cons k v tl -> (k,v) :: list_t_v tl + +let list_t_len (#t : Type0) (ls : list_t t) : nat = length (list_t_v ls) +let list_t_index (#t : Type0) (ls : list_t t) (i : nat{i < list_t_len ls}) : binding t = + index (list_t_v ls) i + +type slot_s (t : Type0) = list (binding t) +type slots_s (t : Type0) = list (slot_s t) + +type slot_t (t : Type0) = list_t t +let slot_t_v #t = list_t_v #t + +/// Representation function for the slots. +let slots_t_v (#t : Type0) (slots : slots_t t) : slots_s t = + map slot_t_v slots + +/// Representation function for the slots, seen as an associative list. +let slots_t_al_v (#t : Type0) (slots : slots_t t) : assoc_list t = + flatten (map list_t_v slots) + +/// High-level type for the hash-map, seen as a list of associative lists (one +/// list per slot). This is the representation we use most, internally. Note that +/// we later introduce a [map_s] representation, which is the one used in the +/// lemmas shown to the user. +type hashMap_s t = list (slot_s t) + +// TODO: why not always have the condition on the length? +// 'nes': "non-empty slots" +type hashMap_s_nes (t : Type0) : Type0 = + hm:hashMap_s t{is_pos_usize (length hm)} + +/// Representation function for [hashMap_t] as a list of slots +let hashMap_t_v (#t : Type0) (hm : hashMap_t t) : hashMap_s t = + map list_t_v hm.slots + +/// Representation function for [hashMap_t] as an associative list +let hashMap_t_al_v (#t : Type0) (hm : hashMap_t t) : assoc_list t = + flatten (hashMap_t_v hm) + +// 'nes': "non-empty slots" +type hashMap_t_nes (t : Type0) : Type0 = + hm:hashMap_t t{is_pos_usize (length hm.slots)} + +let hash_key_s (k : key) : hash = + Return?.v (hash_key k) + +let hash_mod_key (k : key) (len : usize{len > 0}) : hash = + (hash_key_s k) % len + +let not_same_key (#t : Type0) (k : key) (b : binding t) : bool = fst b <> k +let same_key (#t : Type0) (k : key) (b : binding t) : bool = fst b = k + +// We take a [nat] instead of a [hash] on purpose +let same_hash_mod_key (#t : Type0) (len : usize{len > 0}) (h : nat) (b : binding t) : bool = + hash_mod_key (fst b) len = h + +let binding_neq (#t : Type0) (b0 b1 : binding t) : bool = fst b0 <> fst b1 + +let hashMap_t_len_s (#t : Type0) (hm : hashMap_t t) : nat = + hm.num_entries + +let assoc_list_find (#t : Type0) (k : key) (slot : assoc_list t) : option t = + match find (same_key k) slot with + | None -> None + | Some (_, v) -> Some v + +let slot_s_find (#t : Type0) (k : key) (slot : list (binding t)) : option t = + assoc_list_find k slot + +let slot_t_find_s (#t : Type0) (k : key) (slot : list_t t) : option t = + slot_s_find k (slot_t_v slot) + +// This is a simpler version of the "find" function, which captures the essence +// of what happens and operates on [hashMap_s]. +let hashMap_s_find + (#t : Type0) (hm : hashMap_s_nes t) + (k : key) : option t = + let i = hash_mod_key k (length hm) in + let slot = index hm i in + slot_s_find k slot + +let hashMap_s_len + (#t : Type0) (hm : hashMap_s t) : + nat = + length (flatten hm) + +// Same as above, but operates on [hashMap_t] +// Note that we don't reuse the above function on purpose: converting to a +// [hashMap_s] then looking up an element is not the same as what we +// wrote below. +let hashMap_t_find_s + (#t : Type0) (hm : hashMap_t t{length hm.slots > 0}) (k : key) : option t = + let slots = hm.slots in + let i = hash_mod_key k (length slots) in + let slot = index slots i in + slot_t_find_s k slot + +/// Invariants for the slots + +let slot_s_inv + (#t : Type0) (len : usize{len > 0}) (i : usize) (slot : list (binding t)) : bool = + // All the bindings are in the proper slot + for_all (same_hash_mod_key len i) slot && + // All the keys are pairwise distinct + pairwise_rel binding_neq slot + +let slot_t_inv (#t : Type0) (len : usize{len > 0}) (i : usize) (slot : list_t t) : bool = + slot_s_inv len i (slot_t_v slot) + +let slots_s_inv (#t : Type0) (slots : slots_s t{length slots <= usize_max}) : Type0 = + forall(i:nat{i < length slots}). + {:pattern index slots i} + slot_s_inv (length slots) i (index slots i) + +// At some point we tried to rewrite this in terms of [slots_s_inv]. However it +// made a lot of proofs fail because those proofs relied on the [index_map_lem] +// pattern. We tried writing others lemmas with patterns (like [slots_s_inv] +// implies [slots_t_inv]) but it didn't succeed, so we keep things as they are. +let slots_t_inv (#t : Type0) (slots : slots_t t{length slots <= usize_max}) : Type0 = + forall(i:nat{i < length slots}). + {:pattern index slots i} + slot_t_inv (length slots) i (index slots i) + +let hashMap_s_inv (#t : Type0) (hm : hashMap_s t) : Type0 = + length hm <= usize_max /\ + length hm > 0 /\ + slots_s_inv hm + +/// Base invariant for the hashmap (the complete invariant can be temporarily +/// broken between the moment we inserted an element and the moment we resize) +let hashMap_t_base_inv (#t : Type0) (hm : hashMap_t t) : Type0 = + let al = hashMap_t_al_v hm in + // [num_entries] correctly tracks the number of entries in the table + // Note that it gives us that the length of the slots array is <= usize_max: + // [> length <= usize_max + // (because hashMap_num_entries has type `usize`) + hm.num_entries = length al /\ + // Slots invariant + slots_t_inv hm.slots /\ + // The capacity must be > 0 (otherwise we can't resize, because we + // multiply the capacity by two!) + length hm.slots > 0 /\ + // Load computation + begin + let capacity = length hm.slots in + let (dividend, divisor) = hm.max_load_factor in + 0 < dividend /\ dividend < divisor /\ + capacity * dividend >= divisor /\ + hm.max_load = (capacity * dividend) / divisor + end + +/// We often need to frame some values +let hashMap_t_same_params (#t : Type0) (hm0 hm1 : hashMap_t t) : Type0 = + length hm0.slots = length hm1.slots /\ + hm0.max_load = hm1.max_load /\ + hm0.max_load_factor = hm1.max_load_factor + +/// The following invariants, etc. are meant to be revealed to the user through +/// the .fsti. + +/// Invariant for the hashmap +let hashMap_t_inv (#t : Type0) (hm : hashMap_t t) : Type0 = + // Base invariant + hashMap_t_base_inv hm /\ + // The hash map is either: not overloaded, or we can't resize it + begin + let (dividend, divisor) = hm.max_load_factor in + hm.num_entries <= hm.max_load + || length hm.slots * 2 * dividend > usize_max + end + +(*** .fsti *) +/// We reveal slightly different version of the above functions to the user + +let len_s (#t : Type0) (hm : hashMap_t t) : nat = hashMap_t_len_s hm + +/// This version doesn't take any precondition (contrary to [hashMap_t_find_s]) +let find_s (#t : Type0) (hm : hashMap_t t) (k : key) : option t = + if length hm.slots = 0 then None + else hashMap_t_find_s hm k + +(*** Overloading *) + +let hashMap_not_overloaded_lem #t hm = () + +(*** allocate_slots *) + +/// Auxiliary lemma +val slots_t_all_nil_inv_lem + (#t : Type0) (slots : alloc_vec_Vec (list_t t){length slots <= usize_max}) : + Lemma (requires (forall (i:nat{i < length slots}). index slots i == List_Nil)) + (ensures (slots_t_inv slots)) + +#push-options "--fuel 1" +let slots_t_all_nil_inv_lem #t slots = () +#pop-options + +val slots_t_al_v_all_nil_is_empty_lem + (#t : Type0) (slots : alloc_vec_Vec (list_t t)) : + Lemma (requires (forall (i:nat{i < length slots}). index slots i == List_Nil)) + (ensures (slots_t_al_v slots == [])) + +#push-options "--fuel 1" +let rec slots_t_al_v_all_nil_is_empty_lem #t slots = + match slots with + | [] -> () + | s :: slots' -> + assert(forall (i:nat{i < length slots'}). index slots' i == index slots (i+1)); + slots_t_al_v_all_nil_is_empty_lem #t slots'; + assert(slots_t_al_v slots == list_t_v s @ slots_t_al_v slots'); + assert(slots_t_al_v slots == list_t_v s); + assert(index slots 0 == List_Nil) +#pop-options + +/// [allocate_slots] +val hashMap_allocate_slots_lem + (t : Type0) (slots : alloc_vec_Vec (list_t t)) (n : usize) : + Lemma + (requires (length slots + n <= usize_max)) + (ensures ( + match hashMap_allocate_slots t slots n with + | Fail _ -> False + | Return slots' -> + length slots' = length slots + n /\ + // We leave the already allocated slots unchanged + (forall (i:nat{i < length slots}). index slots' i == index slots i) /\ + // We allocate n additional empty slots + (forall (i:nat{length slots <= i /\ i < length slots'}). index slots' i == List_Nil))) + (decreases (hashMap_allocate_slots_loop_decreases t slots n)) + +#push-options "--fuel 1" +let rec hashMap_allocate_slots_lem t slots n = + begin match n with + | 0 -> () + | _ -> + begin match alloc_vec_Vec_push (list_t t) slots List_Nil with + | Fail _ -> () + | Return slots1 -> + begin match usize_sub n 1 with + | Fail _ -> () + | Return i -> + hashMap_allocate_slots_lem t slots1 i; + begin match hashMap_allocate_slots t slots1 i with + | Fail _ -> () + | Return slots2 -> + assert(length slots1 = length slots + 1); + assert(slots1 == slots @ [List_Nil]); // Triggers patterns + assert(index slots1 (length slots) == index [List_Nil] 0); // Triggers patterns + assert(index slots1 (length slots) == List_Nil) + end + end + end + end +#pop-options + +(*** new_with_capacity *) +/// Under proper conditions, [new_with_capacity] doesn't fail and returns an empty hash map. +val hashMap_new_with_capacity_lem + (t : Type0) (capacity : usize) + (max_load_dividend : usize) (max_load_divisor : usize) : + Lemma + (requires ( + 0 < max_load_dividend /\ + max_load_dividend < max_load_divisor /\ + 0 < capacity /\ + capacity * max_load_dividend >= max_load_divisor /\ + capacity * max_load_dividend <= usize_max)) + (ensures ( + match hashMap_new_with_capacity t capacity max_load_dividend max_load_divisor with + | Fail _ -> False + | Return hm -> + // The hash map invariant is satisfied + hashMap_t_inv hm /\ + // The parameters are correct + hm.max_load_factor = (max_load_dividend, max_load_divisor) /\ + hm.max_load = (capacity * max_load_dividend) / max_load_divisor /\ + // The hash map has the specified capacity - we need to reveal this + // otherwise the pre of [hashMap_t_find_s] is not satisfied. + length hm.slots = capacity /\ + // The hash map has 0 values + hashMap_t_len_s hm = 0 /\ + // It contains no bindings + (forall k. hashMap_t_find_s hm k == None) /\ + // We need this low-level property for the invariant + (forall(i:nat{i < length hm.slots}). index hm.slots i == List_Nil))) + +#push-options "--z3rlimit 50 --fuel 1" +let hashMap_new_with_capacity_lem (t : Type0) (capacity : usize) + (max_load_dividend : usize) (max_load_divisor : usize) = + let v = alloc_vec_Vec_new (list_t t) in + assert(length v = 0); + hashMap_allocate_slots_lem t v capacity; + begin match hashMap_allocate_slots t v capacity with + | Fail _ -> assert(False) + | Return v0 -> + begin match usize_mul capacity max_load_dividend with + | Fail _ -> assert(False) + | Return i -> + begin match usize_div i max_load_divisor with + | Fail _ -> assert(False) + | Return i0 -> + let hm = MkhashMap_t 0 (max_load_dividend, max_load_divisor) i0 v0 in + slots_t_all_nil_inv_lem v0; + slots_t_al_v_all_nil_is_empty_lem hm.slots + end + end + end +#pop-options + +(*** new *) + +/// [new] doesn't fail and returns an empty hash map +val hashMap_new_lem_aux (t : Type0) : + Lemma + (ensures ( + match hashMap_new t with + | Fail _ -> False + | Return hm -> + // The hash map invariant is satisfied + hashMap_t_inv hm /\ + // The hash map has 0 values + hashMap_t_len_s hm = 0 /\ + // It contains no bindings + (forall k. hashMap_t_find_s hm k == None))) + +#push-options "--fuel 1" +let hashMap_new_lem_aux t = + hashMap_new_with_capacity_lem t 32 4 5; + match hashMap_new_with_capacity t 32 4 5 with + | Fail _ -> () + | Return hm -> () +#pop-options + +/// The lemma we reveal in the .fsti +let hashMap_new_lem t = hashMap_new_lem_aux t + +(*** clear *) +/// [clear]: the loop doesn't fail and simply clears the slots starting at index i +#push-options "--fuel 1" +let rec hashMap_clear_loop_lem + (t : Type0) (slots : alloc_vec_Vec (list_t t)) (i : usize) : + Lemma + (ensures ( + match hashMap_clear_loop t slots i with + | Fail _ -> False + | Return slots' -> + // The length is preserved + length slots' == length slots /\ + // The slots before i are left unchanged + (forall (j:nat{j < i /\ j < length slots}). index slots' j == index slots j) /\ + // The slots after i are set to List_Nil + (forall (j:nat{i <= j /\ j < length slots}). index slots' j == List_Nil))) + (decreases (hashMap_clear_loop_decreases t slots i)) + = + let i0 = alloc_vec_Vec_len (list_t t) slots in + let b = i < i0 in + if b + then + begin match alloc_vec_Vec_update_usize slots i List_Nil with + | Fail _ -> () + | Return v -> + begin match usize_add i 1 with + | Fail _ -> () + | Return i1 -> + hashMap_clear_loop_lem t v i1; + begin match hashMap_clear_loop t v i1 with + | Fail _ -> () + | Return slots1 -> + assert(length slots1 == length slots); + assert(forall (j:nat{i+1 <= j /\ j < length slots}). index slots1 j == List_Nil); + assert(index slots1 i == List_Nil) + end + end + end + else () +#pop-options + +/// [clear] doesn't fail and turns the hash map into an empty map +val hashMap_clear_lem_aux + (#t : Type0) (self : hashMap_t t) : + Lemma + (requires (hashMap_t_base_inv self)) + (ensures ( + match hashMap_clear t self with + | Fail _ -> False + | Return hm -> + // The hash map invariant is satisfied + hashMap_t_base_inv hm /\ + // We preserved the parameters + hashMap_t_same_params hm self /\ + // The hash map has 0 values + hashMap_t_len_s hm = 0 /\ + // It contains no bindings + (forall k. hashMap_t_find_s hm k == None))) + +// Being lazy: fuel 1 helps a lot... +#push-options "--fuel 1" +let hashMap_clear_lem_aux #t self = + let p = self.max_load_factor in + let i = self.max_load in + let v = self.slots in + hashMap_clear_loop_lem t v 0; + begin match hashMap_clear_loop t v 0 with + | Fail _ -> () + | Return slots1 -> + slots_t_al_v_all_nil_is_empty_lem slots1; + let hm1 = MkhashMap_t 0 p i slots1 in + assert(hashMap_t_base_inv hm1); + assert(hashMap_t_inv hm1) + end +#pop-options + +let hashMap_clear_lem #t self = hashMap_clear_lem_aux #t self + +(*** len *) + +/// [len]: we link it to a non-failing function. +/// Rk.: we might want to make an analysis to not use an error monad to translate +/// functions which statically can't fail. +let hashMap_len_lem #t self = () + + +(*** insert_in_list *) + +(**** insert_in_list'fwd *) + +/// [insert_in_list]: returns true iff the key is not in the list (functional version) +val hashMap_insert_in_list_lem + (t : Type0) (key : usize) (value : t) (ls : list_t t) : + Lemma + (ensures ( + match hashMap_insert_in_list t key value ls with + | Fail _ -> False + | Return b -> + b <==> (slot_t_find_s key ls == None))) + (decreases (hashMap_insert_in_list_loop_decreases t key value ls)) + +#push-options "--fuel 1" +let rec hashMap_insert_in_list_lem t key value ls = + begin match ls with + | List_Cons ckey cvalue ls0 -> + let b = ckey = key in + if b + then () + else + begin + hashMap_insert_in_list_lem t key value ls0; + match hashMap_insert_in_list t key value ls0 with + | Fail _ -> () + | Return b0 -> () + end + | List_Nil -> + assert(list_t_v ls == []); + assert_norm(find (same_key #t key) [] == None) + end +#pop-options + +(**** insert_in_list'back *) + +/// The proofs about [insert_in_list] backward are easier to do in several steps: +/// extrinsic proofs to the rescue! +/// We first prove that [insert_in_list] refines the function we wrote above, then +/// use this function to prove the invariants, etc. + +/// We write a helper which "captures" what [insert_in_list] does. +/// We then reason about this helper to prove the high-level properties we want +/// (functional properties, preservation of invariants, etc.). +let hashMap_insert_in_list_s + (#t : Type0) (key : usize) (value : t) (ls : list (binding t)) : + list (binding t) = + // Check if there is already a binding for the key + match find (same_key key) ls with + | None -> + // No binding: append the binding to the end + ls @ [(key,value)] + | Some _ -> + // There is already a binding: update it + find_update (same_key key) ls (key,value) + +/// [insert_in_list]: if the key is not in the map, appends a new bindings (functional version) +val hashMap_insert_in_list_back_lem_append_s + (t : Type0) (key : usize) (value : t) (ls : list_t t) : + Lemma + (requires ( + slot_t_find_s key ls == None)) + (ensures ( + match hashMap_insert_in_list_back t key value ls with + | Fail _ -> False + | Return ls' -> + list_t_v ls' == list_t_v ls @ [(key,value)])) + (decreases (hashMap_insert_in_list_loop_decreases t key value ls)) + +#push-options "--fuel 1" +let rec hashMap_insert_in_list_back_lem_append_s t key value ls = + begin match ls with + | List_Cons ckey cvalue ls0 -> + let b = ckey = key in + if b + then () + else + begin + hashMap_insert_in_list_back_lem_append_s t key value ls0; + match hashMap_insert_in_list_back t key value ls0 with + | Fail _ -> () + | Return l -> () + end + | List_Nil -> () + end +#pop-options + +/// [insert_in_list]: if the key is in the map, we update the binding (functional version) +val hashMap_insert_in_list_back_lem_update_s + (t : Type0) (key : usize) (value : t) (ls : list_t t) : + Lemma + (requires ( + Some? (find (same_key key) (list_t_v ls)))) + (ensures ( + match hashMap_insert_in_list_back t key value ls with + | Fail _ -> False + | Return ls' -> + list_t_v ls' == find_update (same_key key) (list_t_v ls) (key,value))) + (decreases (hashMap_insert_in_list_loop_decreases t key value ls)) + +#push-options "--fuel 1" +let rec hashMap_insert_in_list_back_lem_update_s t key value ls = + begin match ls with + | List_Cons ckey cvalue ls0 -> + let b = ckey = key in + if b + then () + else + begin + hashMap_insert_in_list_back_lem_update_s t key value ls0; + match hashMap_insert_in_list_back t key value ls0 with + | Fail _ -> () + | Return l -> () + end + | List_Nil -> () + end +#pop-options + +/// Put everything together +val hashMap_insert_in_list_back_lem_s + (t : Type0) (key : usize) (value : t) (ls : list_t t) : + Lemma + (ensures ( + match hashMap_insert_in_list_back t key value ls with + | Fail _ -> False + | Return ls' -> + list_t_v ls' == hashMap_insert_in_list_s key value (list_t_v ls))) + +let hashMap_insert_in_list_back_lem_s t key value ls = + match find (same_key key) (list_t_v ls) with + | None -> hashMap_insert_in_list_back_lem_append_s t key value ls + | Some _ -> hashMap_insert_in_list_back_lem_update_s t key value ls + +(**** Invariants of insert_in_list_s *) + +/// Auxiliary lemmas +/// We work on [hashMap_insert_in_list_s], the "high-level" version of [insert_in_list'back]. +/// +/// Note that in F* we can't have recursive proofs inside of other proofs, contrary +/// to Coq, which makes it a bit cumbersome to prove auxiliary results like the +/// following ones... + +(** Auxiliary lemmas: append case *) + +val slot_t_v_for_all_binding_neq_append_lem + (t : Type0) (key : usize) (value : t) (ls : list (binding t)) (b : binding t) : + Lemma + (requires ( + fst b <> key /\ + for_all (binding_neq b) ls /\ + slot_s_find key ls == None)) + (ensures ( + for_all (binding_neq b) (ls @ [(key,value)]))) + +#push-options "--fuel 1" +let rec slot_t_v_for_all_binding_neq_append_lem t key value ls b = + match ls with + | [] -> () + | (ck, cv) :: cls -> + slot_t_v_for_all_binding_neq_append_lem t key value cls b +#pop-options + +val slot_s_inv_not_find_append_end_inv_lem + (t : Type0) (len : usize{len > 0}) (key : usize) (value : t) (ls : list (binding t)) : + Lemma + (requires ( + slot_s_inv len (hash_mod_key key len) ls /\ + slot_s_find key ls == None)) + (ensures ( + let ls' = ls @ [(key,value)] in + slot_s_inv len (hash_mod_key key len) ls' /\ + (slot_s_find key ls' == Some value) /\ + (forall k'. k' <> key ==> slot_s_find k' ls' == slot_s_find k' ls))) + +#push-options "--fuel 1" +let rec slot_s_inv_not_find_append_end_inv_lem t len key value ls = + match ls with + | [] -> () + | (ck, cv) :: cls -> + slot_s_inv_not_find_append_end_inv_lem t len key value cls; + let h = hash_mod_key key len in + let ls' = ls @ [(key,value)] in + assert(for_all (same_hash_mod_key len h) ls'); + slot_t_v_for_all_binding_neq_append_lem t key value cls (ck, cv); + assert(pairwise_rel binding_neq ls'); + assert(slot_s_inv len h ls') +#pop-options + +/// [insert_in_list]: if the key is not in the map, appends a new bindings +val hashMap_insert_in_list_s_lem_append + (t : Type0) (len : usize{len > 0}) (key : usize) (value : t) (ls : list (binding t)) : + Lemma + (requires ( + slot_s_inv len (hash_mod_key key len) ls /\ + slot_s_find key ls == None)) + (ensures ( + let ls' = hashMap_insert_in_list_s key value ls in + ls' == ls @ [(key,value)] /\ + // The invariant is preserved + slot_s_inv len (hash_mod_key key len) ls' /\ + // [key] maps to [value] + slot_s_find key ls' == Some value /\ + // The other bindings are preserved + (forall k'. k' <> key ==> slot_s_find k' ls' == slot_s_find k' ls))) + +let hashMap_insert_in_list_s_lem_append t len key value ls = + slot_s_inv_not_find_append_end_inv_lem t len key value ls + +/// [insert_in_list]: if the key is not in the map, appends a new bindings (quantifiers) +/// Rk.: we don't use this lemma. +/// TODO: remove? +val hashMap_insert_in_list_back_lem_append + (t : Type0) (len : usize{len > 0}) (key : usize) (value : t) (ls : list_t t) : + Lemma + (requires ( + slot_t_inv len (hash_mod_key key len) ls /\ + slot_t_find_s key ls == None)) + (ensures ( + match hashMap_insert_in_list_back t key value ls with + | Fail _ -> False + | Return ls' -> + list_t_v ls' == list_t_v ls @ [(key,value)] /\ + // The invariant is preserved + slot_t_inv len (hash_mod_key key len) ls' /\ + // [key] maps to [value] + slot_t_find_s key ls' == Some value /\ + // The other bindings are preserved + (forall k'. k' <> key ==> slot_t_find_s k' ls' == slot_t_find_s k' ls))) + +let hashMap_insert_in_list_back_lem_append t len key value ls = + hashMap_insert_in_list_back_lem_s t key value ls; + hashMap_insert_in_list_s_lem_append t len key value (list_t_v ls) + +(** Auxiliary lemmas: update case *) + +val slot_s_find_update_for_all_binding_neq_append_lem + (t : Type0) (key : usize) (value : t) (ls : list (binding t)) (b : binding t) : + Lemma + (requires ( + fst b <> key /\ + for_all (binding_neq b) ls)) + (ensures ( + let ls' = find_update (same_key key) ls (key, value) in + for_all (binding_neq b) ls')) + +#push-options "--fuel 1" +let rec slot_s_find_update_for_all_binding_neq_append_lem t key value ls b = + match ls with + | [] -> () + | (ck, cv) :: cls -> + slot_s_find_update_for_all_binding_neq_append_lem t key value cls b +#pop-options + +/// Annoying auxiliary lemma we have to prove because there is no way to reason +/// properly about closures. +/// I'm really enjoying my time. +val for_all_binding_neq_value_indep + (#t : Type0) (key : key) (v0 v1 : t) (ls : list (binding t)) : + Lemma (for_all (binding_neq (key,v0)) ls = for_all (binding_neq (key,v1)) ls) + +#push-options "--fuel 1" +let rec for_all_binding_neq_value_indep #t key v0 v1 ls = + match ls with + | [] -> () + | _ :: ls' -> for_all_binding_neq_value_indep #t key v0 v1 ls' +#pop-options + +val slot_s_inv_find_append_end_inv_lem + (t : Type0) (len : usize{len > 0}) (key : usize) (value : t) (ls : list (binding t)) : + Lemma + (requires ( + slot_s_inv len (hash_mod_key key len) ls /\ + Some? (slot_s_find key ls))) + (ensures ( + let ls' = find_update (same_key key) ls (key, value) in + slot_s_inv len (hash_mod_key key len) ls' /\ + (slot_s_find key ls' == Some value) /\ + (forall k'. k' <> key ==> slot_s_find k' ls' == slot_s_find k' ls))) + +#push-options "--z3rlimit 50 --fuel 1" +let rec slot_s_inv_find_append_end_inv_lem t len key value ls = + match ls with + | [] -> () + | (ck, cv) :: cls -> + let h = hash_mod_key key len in + let ls' = find_update (same_key key) ls (key, value) in + if ck = key then + begin + assert(ls' == (ck,value) :: cls); + assert(for_all (same_hash_mod_key len h) ls'); + // For pairwise_rel: binding_neq (ck, value) is actually independent + // of `value`. Slightly annoying to prove in F*... + assert(for_all (binding_neq (ck,cv)) cls); + for_all_binding_neq_value_indep key cv value cls; + assert(for_all (binding_neq (ck,value)) cls); + assert(pairwise_rel binding_neq ls'); + assert(slot_s_inv len (hash_mod_key key len) ls') + end + else + begin + slot_s_inv_find_append_end_inv_lem t len key value cls; + assert(for_all (same_hash_mod_key len h) ls'); + slot_s_find_update_for_all_binding_neq_append_lem t key value cls (ck, cv); + assert(pairwise_rel binding_neq ls'); + assert(slot_s_inv len h ls') + end +#pop-options + +/// [insert_in_list]: if the key is in the map, update the bindings +val hashMap_insert_in_list_s_lem_update + (t : Type0) (len : usize{len > 0}) (key : usize) (value : t) (ls : list (binding t)) : + Lemma + (requires ( + slot_s_inv len (hash_mod_key key len) ls /\ + Some? (slot_s_find key ls))) + (ensures ( + let ls' = hashMap_insert_in_list_s key value ls in + ls' == find_update (same_key key) ls (key,value) /\ + // The invariant is preserved + slot_s_inv len (hash_mod_key key len) ls' /\ + // [key] maps to [value] + slot_s_find key ls' == Some value /\ + // The other bindings are preserved + (forall k'. k' <> key ==> slot_s_find k' ls' == slot_s_find k' ls))) + +let hashMap_insert_in_list_s_lem_update t len key value ls = + slot_s_inv_find_append_end_inv_lem t len key value ls + + +/// [insert_in_list]: if the key is in the map, update the bindings +/// TODO: not used: remove? +val hashMap_insert_in_list_back_lem_update + (t : Type0) (len : usize{len > 0}) (key : usize) (value : t) (ls : list_t t) : + Lemma + (requires ( + slot_t_inv len (hash_mod_key key len) ls /\ + Some? (slot_t_find_s key ls))) + (ensures ( + match hashMap_insert_in_list_back t key value ls with + | Fail _ -> False + | Return ls' -> + let als = list_t_v ls in + list_t_v ls' == find_update (same_key key) als (key,value) /\ + // The invariant is preserved + slot_t_inv len (hash_mod_key key len) ls' /\ + // [key] maps to [value] + slot_t_find_s key ls' == Some value /\ + // The other bindings are preserved + (forall k'. k' <> key ==> slot_t_find_s k' ls' == slot_t_find_s k' ls))) + +let hashMap_insert_in_list_back_lem_update t len key value ls = + hashMap_insert_in_list_back_lem_s t key value ls; + hashMap_insert_in_list_s_lem_update t len key value (list_t_v ls) + +(** Final lemmas about [insert_in_list] *) + +/// High-level version +val hashMap_insert_in_list_s_lem + (t : Type0) (len : usize{len > 0}) (key : usize) (value : t) (ls : list (binding t)) : + Lemma + (requires ( + slot_s_inv len (hash_mod_key key len) ls)) + (ensures ( + let ls' = hashMap_insert_in_list_s key value ls in + // The invariant is preserved + slot_s_inv len (hash_mod_key key len) ls' /\ + // [key] maps to [value] + slot_s_find key ls' == Some value /\ + // The other bindings are preserved + (forall k'. k' <> key ==> slot_s_find k' ls' == slot_s_find k' ls) /\ + // The length is incremented, iff we inserted a new key + (match slot_s_find key ls with + | None -> length ls' = length ls + 1 + | Some _ -> length ls' = length ls))) + +let hashMap_insert_in_list_s_lem t len key value ls = + match slot_s_find key ls with + | None -> + assert_norm(length [(key,value)] = 1); + hashMap_insert_in_list_s_lem_append t len key value ls + | Some _ -> + hashMap_insert_in_list_s_lem_update t len key value ls + +/// [insert_in_list] +/// TODO: not used: remove? +val hashMap_insert_in_list_back_lem + (t : Type0) (len : usize{len > 0}) (key : usize) (value : t) (ls : list_t t) : + Lemma + (requires (slot_t_inv len (hash_mod_key key len) ls)) + (ensures ( + match hashMap_insert_in_list_back t key value ls with + | Fail _ -> False + | Return ls' -> + // The invariant is preserved + slot_t_inv len (hash_mod_key key len) ls' /\ + // [key] maps to [value] + slot_t_find_s key ls' == Some value /\ + // The other bindings are preserved + (forall k'. k' <> key ==> slot_t_find_s k' ls' == slot_t_find_s k' ls) /\ + // The length is incremented, iff we inserted a new key + (match slot_t_find_s key ls with + | None -> + list_t_v ls' == list_t_v ls @ [(key,value)] /\ + list_t_len ls' = list_t_len ls + 1 + | Some _ -> + list_t_v ls' == find_update (same_key key) (list_t_v ls) (key,value) /\ + list_t_len ls' = list_t_len ls))) + (decreases (hashMap_insert_in_list_loop_decreases t key value ls)) + +let hashMap_insert_in_list_back_lem t len key value ls = + hashMap_insert_in_list_back_lem_s t key value ls; + hashMap_insert_in_list_s_lem t len key value (list_t_v ls) + +(*** insert_no_resize *) + +(**** Refinement proof *) +/// Same strategy as for [insert_in_list]: we introduce a high-level version of +/// the function, and reason about it. +/// We work on [hashMap_s] (we use a higher-level view of the hash-map, but +/// not too high). + +/// A high-level version of insert, which doesn't check if the table is saturated +let hashMap_insert_no_fail_s + (#t : Type0) (hm : hashMap_s_nes t) + (key : usize) (value : t) : + hashMap_s t = + let len = length hm in + let i = hash_mod_key key len in + let slot = index hm i in + let slot' = hashMap_insert_in_list_s key value slot in + let hm' = list_update hm i slot' in + hm' + +// TODO: at some point I used hashMap_s_nes and it broke proofs...x +let hashMap_insert_no_resize_s + (#t : Type0) (hm : hashMap_s_nes t) + (key : usize) (value : t) : + result (hashMap_s t) = + // Check if the table is saturated (too many entries, and we need to insert one) + let num_entries = length (flatten hm) in + if None? (hashMap_s_find hm key) && num_entries = usize_max then Fail Failure + else Return (hashMap_insert_no_fail_s hm key value) + +/// Prove that [hashMap_insert_no_resize_s] is refined by +/// [hashMap_insert_no_resize'fwd_back] +val hashMap_insert_no_resize_lem_s + (t : Type0) (self : hashMap_t t) (key : usize) (value : t) : + Lemma + (requires ( + hashMap_t_base_inv self /\ + hashMap_s_len (hashMap_t_v self) = hashMap_t_len_s self)) + (ensures ( + begin + match hashMap_insert_no_resize t self key value, + hashMap_insert_no_resize_s (hashMap_t_v self) key value + with + | Fail _, Fail _ -> True + | Return hm, Return hm_v -> + hashMap_t_base_inv hm /\ + hashMap_t_same_params hm self /\ + hashMap_t_v hm == hm_v /\ + hashMap_s_len hm_v == hashMap_t_len_s hm + | _ -> False + end)) + +let hashMap_insert_no_resize_lem_s t self key value = + begin match hash_key key with + | Fail _ -> () + | Return i -> + let i0 = self.num_entries in + let p = self.max_load_factor in + let i1 = self.max_load in + let v = self.slots in + let i2 = alloc_vec_Vec_len (list_t t) v in + let len = length v in + begin match usize_rem i i2 with + | Fail _ -> () + | Return hash_mod -> + begin match alloc_vec_Vec_index_usize v hash_mod with + | Fail _ -> () + | Return l -> + begin + // Checking that: list_t_v (index ...) == index (hashMap_t_v ...) ... + assert(list_t_v l == index (hashMap_t_v self) hash_mod); + hashMap_insert_in_list_lem t key value l; + match hashMap_insert_in_list t key value l with + | Fail _ -> () + | Return b -> + assert(b = None? (slot_s_find key (list_t_v l))); + hashMap_insert_in_list_back_lem t len key value l; + if b + then + begin match usize_add i0 1 with + | Fail _ -> () + | Return i3 -> + begin + match hashMap_insert_in_list_back t key value l with + | Fail _ -> () + | Return l0 -> + begin match alloc_vec_Vec_update_usize v hash_mod l0 with + | Fail _ -> () + | Return v0 -> + let self_v = hashMap_t_v self in + let hm = MkhashMap_t i3 p i1 v0 in + let hm_v = hashMap_t_v hm in + assert(hm_v == list_update self_v hash_mod (list_t_v l0)); + assert_norm(length [(key,value)] = 1); + assert(length (list_t_v l0) = length (list_t_v l) + 1); + length_flatten_update self_v hash_mod (list_t_v l0); + assert(hashMap_s_len hm_v = hashMap_t_len_s hm) + end + end + end + else + begin + match hashMap_insert_in_list_back t key value l with + | Fail _ -> () + | Return l0 -> + begin match alloc_vec_Vec_update_usize v hash_mod l0 with + | Fail _ -> () + | Return v0 -> + let self_v = hashMap_t_v self in + let hm = MkhashMap_t i0 p i1 v0 in + let hm_v = hashMap_t_v hm in + assert(hm_v == list_update self_v hash_mod (list_t_v l0)); + assert(length (list_t_v l0) = length (list_t_v l)); + length_flatten_update self_v hash_mod (list_t_v l0); + assert(hashMap_s_len hm_v = hashMap_t_len_s hm) + end + end + end + end + end + end + +(**** insert_{no_fail,no_resize}: invariants *) + +let hashMap_s_updated_binding + (#t : Type0) (hm : hashMap_s_nes t) + (key : usize) (opt_value : option t) (hm' : hashMap_s_nes t) : Type0 = + // [key] maps to [value] + hashMap_s_find hm' key == opt_value /\ + // The other bindings are preserved + (forall k'. k' <> key ==> hashMap_s_find hm' k' == hashMap_s_find hm k') + +let insert_post (#t : Type0) (hm : hashMap_s_nes t) + (key : usize) (value : t) (hm' : hashMap_s_nes t) : Type0 = + // The invariant is preserved + hashMap_s_inv hm' /\ + // [key] maps to [value] and the other bindings are preserved + hashMap_s_updated_binding hm key (Some value) hm' /\ + // The length is incremented, iff we inserted a new key + (match hashMap_s_find hm key with + | None -> hashMap_s_len hm' = hashMap_s_len hm + 1 + | Some _ -> hashMap_s_len hm' = hashMap_s_len hm) + +val hashMap_insert_no_fail_s_lem + (#t : Type0) (hm : hashMap_s_nes t) + (key : usize) (value : t) : + Lemma + (requires (hashMap_s_inv hm)) + (ensures ( + let hm' = hashMap_insert_no_fail_s hm key value in + insert_post hm key value hm')) + +let hashMap_insert_no_fail_s_lem #t hm key value = + let len = length hm in + let i = hash_mod_key key len in + let slot = index hm i in + hashMap_insert_in_list_s_lem t len key value slot; + let slot' = hashMap_insert_in_list_s key value slot in + length_flatten_update hm i slot' + +val hashMap_insert_no_resize_s_lem + (#t : Type0) (hm : hashMap_s_nes t) + (key : usize) (value : t) : + Lemma + (requires (hashMap_s_inv hm)) + (ensures ( + match hashMap_insert_no_resize_s hm key value with + | Fail _ -> + // Can fail only if we need to create a new binding in + // an already saturated map + hashMap_s_len hm = usize_max /\ + None? (hashMap_s_find hm key) + | Return hm' -> + insert_post hm key value hm')) + +let hashMap_insert_no_resize_s_lem #t hm key value = + let num_entries = length (flatten hm) in + if None? (hashMap_s_find hm key) && num_entries = usize_max then () + else hashMap_insert_no_fail_s_lem hm key value + + +(**** find after insert *) +/// Lemmas about what happens if we call [find] after an insertion + +val hashMap_insert_no_resize_s_get_same_lem + (#t : Type0) (hm : hashMap_s t) + (key : usize) (value : t) : + Lemma (requires (hashMap_s_inv hm)) + (ensures ( + match hashMap_insert_no_resize_s hm key value with + | Fail _ -> True + | Return hm' -> + hashMap_s_find hm' key == Some value)) + +let hashMap_insert_no_resize_s_get_same_lem #t hm key value = + let num_entries = length (flatten hm) in + if None? (hashMap_s_find hm key) && num_entries = usize_max then () + else + begin + let hm' = Return?.v (hashMap_insert_no_resize_s hm key value) in + let len = length hm in + let i = hash_mod_key key len in + let slot = index hm i in + hashMap_insert_in_list_s_lem t len key value slot + end + +val hashMap_insert_no_resize_s_get_diff_lem + (#t : Type0) (hm : hashMap_s t) + (key : usize) (value : t) (key' : usize{key' <> key}) : + Lemma (requires (hashMap_s_inv hm)) + (ensures ( + match hashMap_insert_no_resize_s hm key value with + | Fail _ -> True + | Return hm' -> + hashMap_s_find hm' key' == hashMap_s_find hm key')) + +let hashMap_insert_no_resize_s_get_diff_lem #t hm key value key' = + let num_entries = length (flatten hm) in + if None? (hashMap_s_find hm key) && num_entries = usize_max then () + else + begin + let hm' = Return?.v (hashMap_insert_no_resize_s hm key value) in + let len = length hm in + let i = hash_mod_key key len in + let slot = index hm i in + hashMap_insert_in_list_s_lem t len key value slot; + let i' = hash_mod_key key' len in + if i <> i' then () + else + begin + () + end + end + + +(*** move_elements_from_list *) + +/// Having a great time here: if we use `result (hashMap_s_res t)` as the +/// return type for [hashMap_move_elements_from_list_s] instead of having this +/// awkward match, the proof of [hashMap_move_elements_lem_refin] fails. +/// I guess it comes from F*'s poor subtyping. +/// Followingly, I'm not taking any chance and using [result_hashMap_s] +/// everywhere. +type result_hashMap_s_nes (t : Type0) : Type0 = + res:result (hashMap_s t) { + match res with + | Fail _ -> True + | Return hm -> is_pos_usize (length hm) + } + +let rec hashMap_move_elements_from_list_s + (#t : Type0) (hm : hashMap_s_nes t) + (ls : slot_s t) : + // Do *NOT* use `result (hashMap_s t)` + Tot (result_hashMap_s_nes t) + (decreases ls) = + match ls with + | [] -> Return hm + | (key, value) :: ls' -> + match hashMap_insert_no_resize_s hm key value with + | Fail e -> Fail e + | Return hm' -> + hashMap_move_elements_from_list_s hm' ls' + +/// Refinement lemma +val hashMap_move_elements_from_list_lem + (t : Type0) (ntable : hashMap_t_nes t) (ls : list_t t) : + Lemma (requires (hashMap_t_base_inv ntable)) + (ensures ( + match hashMap_move_elements_from_list t ntable ls, + hashMap_move_elements_from_list_s (hashMap_t_v ntable) (slot_t_v ls) + with + | Fail _, Fail _ -> True + | Return hm', Return hm_v -> + hashMap_t_base_inv hm' /\ + hashMap_t_v hm' == hm_v /\ + hashMap_t_same_params hm' ntable + | _ -> False)) + (decreases (hashMap_move_elements_from_list_loop_decreases t ntable ls)) + +#push-options "--fuel 1" +let rec hashMap_move_elements_from_list_lem t ntable ls = + begin match ls with + | List_Cons k v tl -> + assert(list_t_v ls == (k, v) :: list_t_v tl); + let ls_v = list_t_v ls in + let (_,_) :: tl_v = ls_v in + hashMap_insert_no_resize_lem_s t ntable k v; + begin match hashMap_insert_no_resize t ntable k v with + | Fail _ -> () + | Return h -> + let h_v = Return?.v (hashMap_insert_no_resize_s (hashMap_t_v ntable) k v) in + assert(hashMap_t_v h == h_v); + hashMap_move_elements_from_list_lem t h tl; + begin match hashMap_move_elements_from_list t h tl with + | Fail _ -> () + | Return h0 -> () + end + end + | List_Nil -> () + end +#pop-options + +(*** move_elements *) + +(**** move_elements: refinement 0 *) +/// The proof for [hashMap_move_elements_lem_refin] broke so many times +/// (while it is supposed to be super simple!) that we decided to add one refinement +/// level, to really do things step by step... +/// Doing this refinement layer made me notice that maybe the problem came from +/// the fact that at some point we have to prove `list_t_v List_Nil == []`: I +/// added the corresponding assert to help Z3 and everything became stable. +/// I finally didn't use this "simple" refinement lemma, but I still keep it here +/// because it allows for easy comparisons with [hashMap_move_elements_s]. + +/// [hashMap_move_elements] refines this function, which is actually almost +/// the same (just a little bit shorter and cleaner, and has a pre). +/// +/// The way I wrote the high-level model is the following: +/// - I copy-pasted the definition of [hashMap_move_elements], wrote the +/// signature which links this new definition to [hashMap_move_elements] and +/// checked that the proof passed +/// - I gradually simplified it, while making sure the proof still passes +#push-options "--fuel 1" +let rec hashMap_move_elements_s_simpl + (t : Type0) (ntable : hashMap_t t) + (slots : alloc_vec_Vec (list_t t)) + (i : usize{i <= length slots /\ length slots <= usize_max}) : + Pure (result ((hashMap_t t) & (alloc_vec_Vec (list_t t)))) + (requires (True)) + (ensures (fun res -> + match res, hashMap_move_elements t ntable slots i with + | Fail _, Fail _ -> True + | Return (ntable1, slots1), Return (ntable2, slots2) -> + ntable1 == ntable2 /\ + slots1 == slots2 + | _ -> False)) + (decreases (hashMap_move_elements_loop_decreases t ntable slots i)) + = + if i < length slots + then + let slot = index slots i in + begin match hashMap_move_elements_from_list t ntable slot with + | Fail e -> Fail e + | Return hm' -> + let slots' = list_update slots i List_Nil in + hashMap_move_elements_s_simpl t hm' slots' (i+1) + end + else Return (ntable, slots) +#pop-options + +(**** move_elements: refinement 1 *) +/// We prove a second refinement lemma: calling [move_elements] refines a function +/// which, for every slot, moves the element out of the slot. This first model is +/// almost exactly the translated function, it just uses `list` instead of `list_t`. + +// Note that we ignore the returned slots (we thus don't return a pair: +// only the new hash map in which we moved the elements from the slots): +// this returned value is not used. +let rec hashMap_move_elements_s + (#t : Type0) (hm : hashMap_s_nes t) + (slots : slots_s t) (i : usize{i <= length slots /\ length slots <= usize_max}) : + Tot (result_hashMap_s_nes t) + (decreases (length slots - i)) = + let len = length slots in + if i < len then + begin + let slot = index slots i in + match hashMap_move_elements_from_list_s hm slot with + | Fail e -> Fail e + | Return hm' -> + let slots' = list_update slots i [] in + hashMap_move_elements_s hm' slots' (i+1) + end + else Return hm + +val hashMap_move_elements_lem_refin + (t : Type0) (ntable : hashMap_t t) + (slots : alloc_vec_Vec (list_t t)) (i : usize{i <= length slots}) : + Lemma + (requires ( + hashMap_t_base_inv ntable)) + (ensures ( + match hashMap_move_elements t ntable slots i, + hashMap_move_elements_s (hashMap_t_v ntable) (slots_t_v slots) i + with + | Fail _, Fail _ -> True // We will prove later that this is not possible + | Return (ntable', _), Return ntable'_v -> + hashMap_t_base_inv ntable' /\ + hashMap_t_v ntable' == ntable'_v /\ + hashMap_t_same_params ntable' ntable + | _ -> False)) + (decreases (length slots - i)) + +#restart-solver +#push-options "--fuel 1" +let rec hashMap_move_elements_lem_refin t ntable slots i = + assert(hashMap_t_base_inv ntable); + let i0 = alloc_vec_Vec_len (list_t t) slots in + let b = i < i0 in + if b + then + begin match alloc_vec_Vec_index_usize slots i with + | Fail _ -> () + | Return l -> + let l0 = core_mem_replace (list_t t) l List_Nil in + assert(l0 == l); + hashMap_move_elements_from_list_lem t ntable l0; + begin match hashMap_move_elements_from_list t ntable l0 with + | Fail _ -> () + | Return h -> + let l1 = core_mem_replace_back (list_t t) l List_Nil in + assert(l1 == List_Nil); + assert(slot_t_v #t List_Nil == []); // THIS IS IMPORTANT + begin match alloc_vec_Vec_update_usize slots i l1 with + | Fail _ -> () + | Return v -> + begin match usize_add i 1 with + | Fail _ -> () + | Return i1 -> + hashMap_move_elements_lem_refin t h v i1; + begin match hashMap_move_elements t h v i1 with + | Fail _ -> + assert(Fail? (hashMap_move_elements t ntable slots i)); + () + | Return (ntable', v0) -> () + end + end + end + end + end + else () +#pop-options + + +(**** move_elements: refinement 2 *) +/// We prove a second refinement lemma: calling [move_elements] refines a function +/// which moves every binding of the hash map seen as *one* associative list +/// (and not a list of lists). + +/// [ntable] is the hash map to which we move the elements +/// [slots] is the current hash map, from which we remove the elements, and seen +/// as a "flat" associative list (and not a list of lists) +/// This is actually exactly [hashMap_move_elements_from_list_s]... +let rec hashMap_move_elements_s_flat + (#t : Type0) (ntable : hashMap_s_nes t) + (slots : assoc_list t) : + Tot (result_hashMap_s_nes t) + (decreases slots) = + match slots with + | [] -> Return ntable + | (k,v) :: slots' -> + match hashMap_insert_no_resize_s ntable k v with + | Fail e -> Fail e + | Return ntable' -> + hashMap_move_elements_s_flat ntable' slots' + +/// The refinment lemmas +/// First, auxiliary helpers. + +/// Flatten a list of lists, starting at index i +val flatten_i : + #a:Type + -> l:list (list a) + -> i:nat{i <= length l} + -> Tot (list a) (decreases (length l - i)) + +let rec flatten_i l i = + if i < length l then + index l i @ flatten_i l (i+1) + else [] + +let _ = assert(let l = [1;2] in l == hd l :: tl l) + +val flatten_i_incr : + #a:Type + -> l:list (list a) + -> i:nat{Cons? l /\ i+1 <= length l} -> + Lemma + (ensures ( + (**) assert_norm(length (hd l :: tl l) == 1 + length (tl l)); + flatten_i l (i+1) == flatten_i (tl l) i)) + (decreases (length l - (i+1))) + +#push-options "--fuel 1" +let rec flatten_i_incr l i = + let x :: tl = l in + if i + 1 < length l then + begin + assert(flatten_i l (i+1) == index l (i+1) @ flatten_i l (i+2)); + flatten_i_incr l (i+1); + assert(flatten_i l (i+2) == flatten_i tl (i+1)); + assert(index l (i+1) == index tl i) + end + else () +#pop-options + +val flatten_0_is_flatten : + #a:Type + -> l:list (list a) -> + Lemma + (ensures (flatten_i l 0 == flatten l)) + +#push-options "--fuel 1" +let rec flatten_0_is_flatten #a l = + match l with + | [] -> () + | x :: l' -> + flatten_i_incr l 0; + flatten_0_is_flatten l' +#pop-options + +/// Auxiliary lemma +val flatten_nil_prefix_as_flatten_i : + #a:Type + -> l:list (list a) + -> i:nat{i <= length l} -> + Lemma (requires (forall (j:nat{j < i}). index l j == [])) + (ensures (flatten l == flatten_i l i)) + +#push-options "--fuel 1" +let rec flatten_nil_prefix_as_flatten_i #a l i = + if i = 0 then flatten_0_is_flatten l + else + begin + let x :: l' = l in + assert(index l 0 == []); + assert(x == []); + assert(flatten l == flatten l'); + flatten_i_incr l (i-1); + assert(flatten_i l i == flatten_i l' (i-1)); + assert(forall (j:nat{j < length l'}). index l' j == index l (j+1)); + flatten_nil_prefix_as_flatten_i l' (i-1); + assert(flatten l' == flatten_i l' (i-1)) + end +#pop-options + +/// The proof is trivial, the functions are the same. +/// Just keeping two definitions to allow changes... +val hashMap_move_elements_from_list_s_as_flat_lem + (#t : Type0) (hm : hashMap_s_nes t) + (ls : slot_s t) : + Lemma + (ensures ( + hashMap_move_elements_from_list_s hm ls == + hashMap_move_elements_s_flat hm ls)) + (decreases ls) + +#push-options "--fuel 1" +let rec hashMap_move_elements_from_list_s_as_flat_lem #t hm ls = + match ls with + | [] -> () + | (key, value) :: ls' -> + match hashMap_insert_no_resize_s hm key value with + | Fail _ -> () + | Return hm' -> + hashMap_move_elements_from_list_s_as_flat_lem hm' ls' +#pop-options + +/// Composition of two calls to [hashMap_move_elements_s_flat] +let hashMap_move_elements_s_flat_comp + (#t : Type0) (hm : hashMap_s_nes t) (slot0 slot1 : slot_s t) : + Tot (result_hashMap_s_nes t) = + match hashMap_move_elements_s_flat hm slot0 with + | Fail e -> Fail e + | Return hm1 -> hashMap_move_elements_s_flat hm1 slot1 + +/// High-level desc: +/// move_elements (move_elements hm slot0) slo1 == move_elements hm (slot0 @ slot1) +val hashMap_move_elements_s_flat_append_lem + (#t : Type0) (hm : hashMap_s_nes t) (slot0 slot1 : slot_s t) : + Lemma + (ensures ( + match hashMap_move_elements_s_flat_comp hm slot0 slot1, + hashMap_move_elements_s_flat hm (slot0 @ slot1) + with + | Fail _, Fail _ -> True + | Return hm1, Return hm2 -> hm1 == hm2 + | _ -> False)) + (decreases (slot0)) + +#push-options "--fuel 1" +let rec hashMap_move_elements_s_flat_append_lem #t hm slot0 slot1 = + match slot0 with + | [] -> () + | (k,v) :: slot0' -> + match hashMap_insert_no_resize_s hm k v with + | Fail _ -> () + | Return hm' -> + hashMap_move_elements_s_flat_append_lem hm' slot0' slot1 +#pop-options + +val flatten_i_same_suffix (#a : Type) (l0 l1 : list (list a)) (i : nat) : + Lemma + (requires ( + i <= length l0 /\ + length l0 = length l1 /\ + (forall (j:nat{i <= j /\ j < length l0}). index l0 j == index l1 j))) + (ensures (flatten_i l0 i == flatten_i l1 i)) + (decreases (length l0 - i)) + +#push-options "--fuel 1" +let rec flatten_i_same_suffix #a l0 l1 i = + if i < length l0 then + flatten_i_same_suffix l0 l1 (i+1) + else () +#pop-options + +/// Refinement lemma: +/// [hashMap_move_elements_s] refines [hashMap_move_elements_s_flat] +/// (actually the functions are equal on all inputs). +val hashMap_move_elements_s_lem_refin_flat + (#t : Type0) (hm : hashMap_s_nes t) + (slots : slots_s t) + (i : nat{i <= length slots /\ length slots <= usize_max}) : + Lemma + (ensures ( + match hashMap_move_elements_s hm slots i, + hashMap_move_elements_s_flat hm (flatten_i slots i) + with + | Fail _, Fail _ -> True + | Return hm, Return hm' -> hm == hm' + | _ -> False)) + (decreases (length slots - i)) + +#push-options "--fuel 1" +let rec hashMap_move_elements_s_lem_refin_flat #t hm slots i = + let len = length slots in + if i < len then + begin + let slot = index slots i in + hashMap_move_elements_from_list_s_as_flat_lem hm slot; + match hashMap_move_elements_from_list_s hm slot with + | Fail _ -> + assert(flatten_i slots i == slot @ flatten_i slots (i+1)); + hashMap_move_elements_s_flat_append_lem hm slot (flatten_i slots (i+1)); + assert(Fail? (hashMap_move_elements_s_flat hm (flatten_i slots i))) + | Return hm' -> + let slots' = list_update slots i [] in + flatten_i_same_suffix slots slots' (i+1); + hashMap_move_elements_s_lem_refin_flat hm' slots' (i+1); + hashMap_move_elements_s_flat_append_lem hm slot (flatten_i slots' (i+1)); + () + end + else () +#pop-options + +let assoc_list_inv (#t : Type0) (al : assoc_list t) : Type0 = + // All the keys are pairwise distinct + pairwise_rel binding_neq al + +let disjoint_hm_al_on_key + (#t : Type0) (hm : hashMap_s_nes t) (al : assoc_list t) (k : key) : Type0 = + match hashMap_s_find hm k, assoc_list_find k al with + | Some _, None + | None, Some _ + | None, None -> True + | Some _, Some _ -> False + +/// Playing a dangerous game here: using forall quantifiers +let disjoint_hm_al (#t : Type0) (hm : hashMap_s_nes t) (al : assoc_list t) : Type0 = + forall (k:key). disjoint_hm_al_on_key hm al k + +let find_in_union_hm_al + (#t : Type0) (hm : hashMap_s_nes t) (al : assoc_list t) (k : key) : + option t = + match hashMap_s_find hm k with + | Some b -> Some b + | None -> assoc_list_find k al + +/// Auxiliary lemma +val for_all_binding_neq_find_lem (#t : Type0) (k : key) (v : t) (al : assoc_list t) : + Lemma (requires (for_all (binding_neq (k,v)) al)) + (ensures (assoc_list_find k al == None)) + +#push-options "--fuel 1" +let rec for_all_binding_neq_find_lem #t k v al = + match al with + | [] -> () + | b :: al' -> for_all_binding_neq_find_lem k v al' +#pop-options + +val hashMap_move_elements_s_flat_lem + (#t : Type0) (hm : hashMap_s_nes t) (al : assoc_list t) : + Lemma + (requires ( + // Invariants + hashMap_s_inv hm /\ + assoc_list_inv al /\ + // The two are disjoint + disjoint_hm_al hm al /\ + // We can add all the elements to the hashmap + hashMap_s_len hm + length al <= usize_max)) + (ensures ( + match hashMap_move_elements_s_flat hm al with + | Fail _ -> False // We can't fail + | Return hm' -> + // The invariant is preserved + hashMap_s_inv hm' /\ + // The new hash map is the union of the two maps + (forall (k:key). hashMap_s_find hm' k == find_in_union_hm_al hm al k) /\ + hashMap_s_len hm' = hashMap_s_len hm + length al)) + (decreases al) + +#restart-solver +#push-options "--z3rlimit 200 --fuel 1" +let rec hashMap_move_elements_s_flat_lem #t hm al = + match al with + | [] -> () + | (k,v) :: al' -> + hashMap_insert_no_resize_s_lem hm k v; + match hashMap_insert_no_resize_s hm k v with + | Fail _ -> () + | Return hm' -> + assert(hashMap_s_inv hm'); + assert(assoc_list_inv al'); + let disjoint_lem (k' : key) : + Lemma (disjoint_hm_al_on_key hm' al' k') + [SMTPat (disjoint_hm_al_on_key hm' al' k')] = + if k' = k then + begin + assert(hashMap_s_find hm' k' == Some v); + for_all_binding_neq_find_lem k v al'; + assert(assoc_list_find k' al' == None) + end + else + begin + assert(hashMap_s_find hm' k' == hashMap_s_find hm k'); + assert(assoc_list_find k' al' == assoc_list_find k' al) + end + in + assert(disjoint_hm_al hm' al'); + assert(hashMap_s_len hm' + length al' <= usize_max); + hashMap_move_elements_s_flat_lem hm' al' +#pop-options + +/// We need to prove that the invariants on the "low-level" representations of +/// the hash map imply the invariants on the "high-level" representations. + +val slots_t_inv_implies_slots_s_inv + (#t : Type0) (slots : slots_t t{length slots <= usize_max}) : + Lemma (requires (slots_t_inv slots)) + (ensures (slots_s_inv (slots_t_v slots))) + +let slots_t_inv_implies_slots_s_inv #t slots = + // Ok, works fine: this lemma was useless. + // Problem is: I can never really predict for sure with F*... + () + +val hashMap_t_base_inv_implies_hashMap_s_inv + (#t : Type0) (hm : hashMap_t t) : + Lemma (requires (hashMap_t_base_inv hm)) + (ensures (hashMap_s_inv (hashMap_t_v hm))) + +let hashMap_t_base_inv_implies_hashMap_s_inv #t hm = () // same as previous + +/// Introducing a "partial" version of the hash map invariant, which operates on +/// a suffix of the hash map. +let partial_hashMap_s_inv + (#t : Type0) (len : usize{len > 0}) (offset : usize) + (hm : hashMap_s t{offset + length hm <= usize_max}) : Type0 = + forall(i:nat{i < length hm}). {:pattern index hm i} slot_s_inv len (offset + i) (index hm i) + +/// Auxiliary lemma. +/// If a binding comes from a slot i, then its key is different from the keys +/// of the bindings in the other slots (because the hashes of the keys are distinct). +val binding_in_previous_slot_implies_neq + (#t : Type0) (len : usize{len > 0}) + (i : usize) (b : binding t) + (offset : usize{i < offset}) + (slots : hashMap_s t{offset + length slots <= usize_max}) : + Lemma + (requires ( + // The binding comes from a slot not in [slots] + hash_mod_key (fst b) len = i /\ + // The slots are the well-formed suffix of a hash map + partial_hashMap_s_inv len offset slots)) + (ensures ( + for_all (binding_neq b) (flatten slots))) + (decreases slots) + +#push-options "--z3rlimit 100 --fuel 1" +let rec binding_in_previous_slot_implies_neq #t len i b offset slots = + match slots with + | [] -> () + | s :: slots' -> + assert(slot_s_inv len offset (index slots 0)); // Triggers patterns + assert(slot_s_inv len offset s); + // Proving TARGET. We use quantifiers. + assert(for_all (same_hash_mod_key len offset) s); + forall_index_equiv_list_for_all (same_hash_mod_key len offset) s; + assert(forall (i:nat{i < length s}). same_hash_mod_key len offset (index s i)); + let aux (i:nat{i < length s}) : + Lemma + (requires (same_hash_mod_key len offset (index s i))) + (ensures (binding_neq b (index s i))) + [SMTPat (index s i)] = () + in + assert(forall (i:nat{i < length s}). binding_neq b (index s i)); + forall_index_equiv_list_for_all (binding_neq b) s; + assert(for_all (binding_neq b) s); // TARGET + // + assert(forall (i:nat{i < length slots'}). index slots' i == index slots (i+1)); // Triggers instantiations + binding_in_previous_slot_implies_neq len i b (offset+1) slots'; + for_all_append (binding_neq b) s (flatten slots') +#pop-options + +val partial_hashMap_s_inv_implies_assoc_list_lem + (#t : Type0) (len : usize{len > 0}) (offset : usize) + (hm : hashMap_s t{offset + length hm <= usize_max}) : + Lemma + (requires ( + partial_hashMap_s_inv len offset hm)) + (ensures (assoc_list_inv (flatten hm))) + (decreases (length hm + length (flatten hm))) + +#push-options "--fuel 1" +let rec partial_hashMap_s_inv_implies_assoc_list_lem #t len offset hm = + match hm with + | [] -> () + | slot :: hm' -> + assert(flatten hm == slot @ flatten hm'); + assert(forall (i:nat{i < length hm'}). index hm' i == index hm (i+1)); // Triggers instantiations + match slot with + | [] -> + assert(flatten hm == flatten hm'); + assert(partial_hashMap_s_inv len (offset+1) hm'); // Triggers instantiations + partial_hashMap_s_inv_implies_assoc_list_lem len (offset+1) hm' + | x :: slot' -> + assert(flatten (slot' :: hm') == slot' @ flatten hm'); + let hm'' = slot' :: hm' in + assert(forall (i:nat{0 < i /\ i < length hm''}). index hm'' i == index hm i); // Triggers instantiations + assert(forall (i:nat{0 < i /\ i < length hm''}). slot_s_inv len (offset + i) (index hm'' i)); + assert(index hm 0 == slot); // Triggers instantiations + assert(slot_s_inv len offset slot); + assert(slot_s_inv len offset slot'); + assert(partial_hashMap_s_inv len offset hm''); + partial_hashMap_s_inv_implies_assoc_list_lem len offset (slot' :: hm'); + // Proving that the key in `x` is different from all the other keys in + // the flattened map + assert(for_all (binding_neq x) slot'); + for_all_append (binding_neq x) slot' (flatten hm'); + assert(partial_hashMap_s_inv len (offset+1) hm'); + binding_in_previous_slot_implies_neq #t len offset x (offset+1) hm'; + assert(for_all (binding_neq x) (flatten hm')); + assert(for_all (binding_neq x) (flatten (slot' :: hm'))) +#pop-options + +val hashMap_s_inv_implies_assoc_list_lem + (#t : Type0) (hm : hashMap_s t) : + Lemma (requires (hashMap_s_inv hm)) + (ensures (assoc_list_inv (flatten hm))) + +let hashMap_s_inv_implies_assoc_list_lem #t hm = + partial_hashMap_s_inv_implies_assoc_list_lem (length hm) 0 hm + +val hashMap_t_base_inv_implies_assoc_list_lem + (#t : Type0) (hm : hashMap_t t): + Lemma (requires (hashMap_t_base_inv hm)) + (ensures (assoc_list_inv (hashMap_t_al_v hm))) + +let hashMap_t_base_inv_implies_assoc_list_lem #t hm = + hashMap_s_inv_implies_assoc_list_lem (hashMap_t_v hm) + +/// For some reason, we can't write the below [forall] directly in the [ensures] +/// clause of the next lemma: it makes Z3 fails even with a huge rlimit. +/// I have no idea what's going on. +let hashMap_is_assoc_list + (#t : Type0) (ntable : hashMap_t t{length ntable.slots > 0}) + (al : assoc_list t) : Type0 = + (forall (k:key). hashMap_t_find_s ntable k == assoc_list_find k al) + +let partial_hashMap_s_find + (#t : Type0) (len : usize{len > 0}) (offset : usize) + (hm : hashMap_s_nes t{offset + length hm = len}) + (k : key{hash_mod_key k len >= offset}) : option t = + let i = hash_mod_key k len in + let slot = index hm (i - offset) in + slot_s_find k slot + +val not_same_hash_key_not_found_in_slot + (#t : Type0) (len : usize{len > 0}) + (k : key) + (i : usize) + (slot : slot_s t) : + Lemma + (requires ( + hash_mod_key k len <> i /\ + slot_s_inv len i slot)) + (ensures (slot_s_find k slot == None)) + +#push-options "--fuel 1" +let rec not_same_hash_key_not_found_in_slot #t len k i slot = + match slot with + | [] -> () + | (k',v) :: slot' -> not_same_hash_key_not_found_in_slot len k i slot' +#pop-options + +/// Small variation of [binding_in_previous_slot_implies_neq]: if the hash of +/// a key links it to a previous slot, it can't be found in the slots after. +val key_in_previous_slot_implies_not_found + (#t : Type0) (len : usize{len > 0}) + (k : key) + (offset : usize) + (slots : hashMap_s t{offset + length slots = len}) : + Lemma + (requires ( + // The binding comes from a slot not in [slots] + hash_mod_key k len < offset /\ + // The slots are the well-formed suffix of a hash map + partial_hashMap_s_inv len offset slots)) + (ensures ( + assoc_list_find k (flatten slots) == None)) + (decreases slots) + +#push-options "--fuel 1" +let rec key_in_previous_slot_implies_not_found #t len k offset slots = + match slots with + | [] -> () + | slot :: slots' -> + find_append (same_key k) slot (flatten slots'); + assert(index slots 0 == slot); // Triggers instantiations + not_same_hash_key_not_found_in_slot #t len k offset slot; + assert(assoc_list_find k slot == None); + assert(forall (i:nat{i < length slots'}). index slots' i == index slots (i+1)); // Triggers instantiations + key_in_previous_slot_implies_not_found len k (offset+1) slots' +#pop-options + +val partial_hashMap_s_is_assoc_list_lem + (#t : Type0) (len : usize{len > 0}) (offset : usize) + (hm : hashMap_s_nes t{offset + length hm = len}) + (k : key{hash_mod_key k len >= offset}) : + Lemma + (requires ( + partial_hashMap_s_inv len offset hm)) + (ensures ( + partial_hashMap_s_find len offset hm k == assoc_list_find k (flatten hm))) + (decreases hm) + +#push-options "--fuel 1" +let rec partial_hashMap_s_is_assoc_list_lem #t len offset hm k = + match hm with + | [] -> () + | slot :: hm' -> + let h = hash_mod_key k len in + let i = h - offset in + if i = 0 then + begin + // We must look in the current slot + assert(partial_hashMap_s_find len offset hm k == slot_s_find k slot); + find_append (same_key k) slot (flatten hm'); + assert(forall (i:nat{i < length hm'}). index hm' i == index hm (i+1)); // Triggers instantiations + key_in_previous_slot_implies_not_found #t len k (offset+1) hm'; + assert( // Of course, writing `== None` doesn't work... + match find (same_key k) (flatten hm') with + | None -> True + | Some _ -> False); + assert( + find (same_key k) (flatten hm) == + begin match find (same_key k) slot with + | Some x -> Some x + | None -> find (same_key k) (flatten hm') + end); + () + end + else + begin + // We must ignore the current slot + assert(partial_hashMap_s_find len offset hm k == + partial_hashMap_s_find len (offset+1) hm' k); + find_append (same_key k) slot (flatten hm'); + assert(index hm 0 == slot); // Triggers instantiations + not_same_hash_key_not_found_in_slot #t len k offset slot; + assert(forall (i:nat{i < length hm'}). index hm' i == index hm (i+1)); // Triggers instantiations + partial_hashMap_s_is_assoc_list_lem #t len (offset+1) hm' k + end +#pop-options + +val hashMap_is_assoc_list_lem (#t : Type0) (hm : hashMap_t t) : + Lemma (requires (hashMap_t_base_inv hm)) + (ensures (hashMap_is_assoc_list hm (hashMap_t_al_v hm))) + +let hashMap_is_assoc_list_lem #t hm = + let aux (k:key) : + Lemma (hashMap_t_find_s hm k == assoc_list_find k (hashMap_t_al_v hm)) + [SMTPat (hashMap_t_find_s hm k)] = + let hm_v = hashMap_t_v hm in + let len = length hm_v in + partial_hashMap_s_is_assoc_list_lem #t len 0 hm_v k + in + () + +/// The final lemma about [move_elements]: calling it on an empty hash table moves +/// all the elements to this empty table. +val hashMap_move_elements_lem + (t : Type0) (ntable : hashMap_t t) (slots : alloc_vec_Vec (list_t t)) : + Lemma + (requires ( + let al = flatten (slots_t_v slots) in + hashMap_t_base_inv ntable /\ + length al <= usize_max /\ + assoc_list_inv al /\ + // The table is empty + hashMap_t_len_s ntable = 0 /\ + (forall (k:key). hashMap_t_find_s ntable k == None))) + (ensures ( + let al = flatten (slots_t_v slots) in + match hashMap_move_elements t ntable slots 0, + hashMap_move_elements_s_flat (hashMap_t_v ntable) al + with + | Return (ntable', _), Return ntable'_v -> + // The invariant is preserved + hashMap_t_base_inv ntable' /\ + // We preserved the parameters + hashMap_t_same_params ntable' ntable /\ + // The table has the same number of slots + length ntable'.slots = length ntable.slots /\ + // The count is good + hashMap_t_len_s ntable' = length al /\ + // The table can be linked to its model (we need this only to reveal + // "pretty" functional lemmas to the user in the fsti - so that we + // can write lemmas with SMT patterns - this is very F* specific) + hashMap_t_v ntable' == ntable'_v /\ + // The new table contains exactly all the bindings from the slots + // Rk.: see the comment for [hashMap_is_assoc_list] + hashMap_is_assoc_list ntable' al + | _ -> False // We can only succeed + )) + +// Weird, dirty things happen below. +// Manually unfolding some postconditions allowed to make the proof pass, +// and also revealed the reason why some proofs failed with "Unknown assertion +// failed" (resulting in the call to [flatten_0_is_flatten] for instance). +// I think manually unfolding the postconditions allowed to account for the +// lack of ifuel (this kind of proofs is annoying, really). +#restart-solver +#push-options "--z3rlimit 100" +let hashMap_move_elements_lem t ntable slots = + let ntable_v = hashMap_t_v ntable in + let slots_v = slots_t_v slots in + let al = flatten slots_v in + hashMap_move_elements_lem_refin t ntable slots 0; + begin + match hashMap_move_elements t ntable slots 0, + hashMap_move_elements_s ntable_v slots_v 0 + with + | Fail _, Fail _ -> () + | Return (ntable', _), Return ntable'_v -> + assert(hashMap_t_base_inv ntable'); + assert(hashMap_t_v ntable' == ntable'_v) + | _ -> assert(False) + end; + hashMap_move_elements_s_lem_refin_flat ntable_v slots_v 0; + begin + match hashMap_move_elements_s ntable_v slots_v 0, + hashMap_move_elements_s_flat ntable_v (flatten_i slots_v 0) + with + | Fail _, Fail _ -> () + | Return hm, Return hm' -> assert(hm == hm') + | _ -> assert(False) + end; + flatten_0_is_flatten slots_v; // flatten_i slots_v 0 == flatten slots_v + hashMap_move_elements_s_flat_lem ntable_v al; + match hashMap_move_elements t ntable slots 0, + hashMap_move_elements_s_flat ntable_v al + with + | Return (ntable', _), Return ntable'_v -> + assert(hashMap_t_base_inv ntable'); + assert(length ntable'.slots = length ntable.slots); + assert(hashMap_t_len_s ntable' = length al); + assert(hashMap_t_v ntable' == ntable'_v); + assert(hashMap_is_assoc_list ntable' al) + | _ -> assert(False) +#pop-options + +(*** try_resize *) + +/// High-level model 1. +/// This is one is slightly "crude": we just simplify a bit the function. + +let hashMap_try_resize_s_simpl + (#t : Type0) + (hm : hashMap_t t) : + Pure (result (hashMap_t t)) + (requires ( + let (divid, divis) = hm.max_load_factor in + divid > 0 /\ divis > 0)) + (ensures (fun _ -> True)) = + let capacity = length hm.slots in + let (divid, divis) = hm.max_load_factor in + if capacity <= (usize_max / 2) / divid then + let ncapacity : usize = capacity * 2 in + begin match hashMap_new_with_capacity t ncapacity divid divis with + | Fail e -> Fail e + | Return ntable -> + match hashMap_move_elements t ntable hm.slots 0 with + | Fail e -> Fail e + | Return (ntable', _) -> + let hm = + { hm with slots = ntable'.slots; + max_load = ntable'.max_load } + in + Return hm + end + else Return hm + +val hashMap_try_resize_lem_refin + (t : Type0) (self : hashMap_t t) : + Lemma + (requires ( + let (divid, divis) = self.max_load_factor in + divid > 0 /\ divis > 0)) + (ensures ( + match hashMap_try_resize t self, + hashMap_try_resize_s_simpl self + with + | Fail _, Fail _ -> True + | Return hm1, Return hm2 -> hm1 == hm2 + | _ -> False)) + +let hashMap_try_resize_lem_refin t self = () + +/// Isolating arithmetic proofs + +let gt_lem0 (n m q : nat) : + Lemma (requires (m > 0 /\ n > q)) + (ensures (n * m > q * m)) = () + +let ge_lem0 (n m q : nat) : + Lemma (requires (m > 0 /\ n >= q)) + (ensures (n * m >= q * m)) = () + +let gt_ge_trans (n m p : nat) : + Lemma (requires (n > m /\ m >= p)) (ensures (n > p)) = () + +let ge_trans (n m p : nat) : + Lemma (requires (n >= m /\ m >= p)) (ensures (n >= p)) = () + +#push-options "--z3rlimit 200" +let gt_lem1 (n m q : nat) : + Lemma (requires (m > 0 /\ n > q / m)) (ensures (n * m > q)) = + assert(n >= q / m + 1); + ge_lem0 n m (q / m + 1); + assert(n * m >= (q / m) * m + m) +#pop-options + +let gt_lem2 (n m p q : nat) : + Lemma (requires (m > 0 /\ p > 0 /\ n > (q / m) / p)) (ensures (n * m * p > q)) = + gt_lem1 n p (q / m); + assert(n * p > q / m); + gt_lem1 (n * p) m q + +let ge_lem1 (n m q : nat) : + Lemma (requires (n >= m /\ q > 0)) + (ensures (n / q >= m / q)) = + FStar.Math.Lemmas.lemma_div_le m n q + +#restart-solver +#push-options "--z3rlimit 200" +let times_divid_lem (n m p : pos) : Lemma ((n * m) / p >= n * (m / p)) + = + FStar.Math.Lemmas.multiply_fractions m p; + assert(m >= (m / p) * p); + assert(n * m >= n * (m / p) * p); // + ge_lem1 (n * m) (n * (m / p) * p) p; + assert((n * m) / p >= (n * (m / p) * p) / p); + assert(n * (m / p) * p = (n * (m / p)) * p); + FStar.Math.Lemmas.cancel_mul_div (n * (m / p)) p; + assert(((n * (m / p)) * p) / p = n * (m / p)) +#pop-options + +/// The good old arithmetic proofs and their unstability... +/// At some point I thought it was stable because it worked with `--quake 100`. +/// Of course, it broke the next time I checked the file... +/// It seems things are ok when we check this proof on its own, but not when +/// it is sent at the same time as the one above (though we put #restart-solver!). +/// I also tried `--quake 1/100` to no avail: it seems that when Z3 decides to +/// fail the first one, it fails them all. I inserted #restart-solver before +/// the previous lemma to see if it had an effect (of course not). +val new_max_load_lem + (len : usize) (capacity : usize{capacity > 0}) + (divid : usize{divid > 0}) (divis : usize{divis > 0}) : + Lemma + (requires ( + let max_load = (capacity * divid) / divis in + let ncapacity = 2 * capacity in + let nmax_load = (ncapacity * divid) / divis in + capacity > 0 /\ 0 < divid /\ divid < divis /\ + capacity * divid >= divis /\ + len = max_load + 1)) + (ensures ( + let max_load = (capacity * divid) / divis in + let ncapacity = 2 * capacity in + let nmax_load = (ncapacity * divid) / divis in + len <= nmax_load)) + +let mul_assoc (a b c : nat) : Lemma (a * b * c == a * (b * c)) = () + +let ge_lem2 (a b c d : nat) : Lemma (requires (a >= b + c /\ c >= d)) (ensures (a >= b + d)) = () +let ge_div_lem1 (a b : nat) : Lemma (requires (a >= b /\ b > 0)) (ensures (a / b >= 1)) = () + +#restart-solver +#push-options "--z3rlimit 100 --z3cliopt smt.arith.nl=false" +let new_max_load_lem len capacity divid divis = + FStar.Math.Lemmas.paren_mul_left 2 capacity divid; + mul_assoc 2 capacity divid; + // The following assertion often breaks though it is given by the above + // lemma. I really don't know what to do (I deactivated non-linear + // arithmetic and added the previous lemma call, moved the assertion up, + // boosted the rlimit...). + assert(2 * capacity * divid == 2 * (capacity * divid)); + let max_load = (capacity * divid) / divis in + let ncapacity = 2 * capacity in + let nmax_load = (ncapacity * divid) / divis in + assert(nmax_load = (2 * capacity * divid) / divis); + times_divid_lem 2 (capacity * divid) divis; + assert((2 * (capacity * divid)) / divis >= 2 * ((capacity * divid) / divis)); + assert(nmax_load >= 2 * ((capacity * divid) / divis)); + assert(nmax_load >= 2 * max_load); + assert(nmax_load >= max_load + max_load); + ge_div_lem1 (capacity * divid) divis; + ge_lem2 nmax_load max_load max_load 1; + assert(nmax_load >= max_load + 1) +#pop-options + +val hashMap_try_resize_s_simpl_lem (#t : Type0) (hm : hashMap_t t) : + Lemma + (requires ( + // The base invariant is satisfied + hashMap_t_base_inv hm /\ + // However, the "full" invariant is broken, as we call [try_resize] + // only if the current number of entries is > the max load. + // + // There are two situations: + // - either we just reached the max load + // - or we were already saturated and can't resize + (let (dividend, divisor) = hm.max_load_factor in + hm.num_entries == hm.max_load + 1 \/ + length hm.slots * 2 * dividend > usize_max) + )) + (ensures ( + match hashMap_try_resize_s_simpl hm with + | Fail _ -> False + | Return hm' -> + // The full invariant is now satisfied (the full invariant is "base + // invariant" + the map is not overloaded (or can't be resized because + // already too big) + hashMap_t_inv hm' /\ + // It contains the same bindings as the initial map + (forall (k:key). hashMap_t_find_s hm' k == hashMap_t_find_s hm k))) + +#restart-solver +#push-options "--z3rlimit 400" +let hashMap_try_resize_s_simpl_lem #t hm = + let capacity = length hm.slots in + let (divid, divis) = hm.max_load_factor in + if capacity <= (usize_max / 2) / divid then + begin + let ncapacity : usize = capacity * 2 in + assert(ncapacity * divid <= usize_max); + assert(hashMap_t_len_s hm = hm.max_load + 1); + new_max_load_lem (hashMap_t_len_s hm) capacity divid divis; + hashMap_new_with_capacity_lem t ncapacity divid divis; + match hashMap_new_with_capacity t ncapacity divid divis with + | Fail _ -> () + | Return ntable -> + let slots = hm.slots in + let al = flatten (slots_t_v slots) in + // Proving that: length al = hm.num_entries + assert(al == flatten (map slot_t_v slots)); + assert(al == flatten (map list_t_v slots)); + assert(hashMap_t_al_v hm == flatten (hashMap_t_v hm)); + assert(hashMap_t_al_v hm == flatten (map list_t_v hm.slots)); + assert(al == hashMap_t_al_v hm); + assert(hashMap_t_base_inv ntable); + assert(length al = hm.num_entries); + assert(length al <= usize_max); + hashMap_t_base_inv_implies_assoc_list_lem hm; + assert(assoc_list_inv al); + assert(hashMap_t_len_s ntable = 0); + assert(forall (k:key). hashMap_t_find_s ntable k == None); + hashMap_move_elements_lem t ntable hm.slots; + match hashMap_move_elements t ntable hm.slots 0 with + | Fail _ -> () + | Return (ntable', _) -> + hashMap_is_assoc_list_lem hm; + assert(hashMap_is_assoc_list hm (hashMap_t_al_v hm)); + let hm' = + { hm with slots = ntable'.slots; + max_load = ntable'.max_load } + in + assert(hashMap_t_base_inv ntable'); + assert(hashMap_t_base_inv hm'); + assert(hashMap_t_len_s hm' = hashMap_t_len_s hm); + new_max_load_lem (hashMap_t_len_s hm') capacity divid divis; + assert(hashMap_t_len_s hm' <= hm'.max_load); // Requires a lemma + assert(hashMap_t_inv hm') + end + else + begin + gt_lem2 capacity 2 divid usize_max; + assert(capacity * 2 * divid > usize_max) + end +#pop-options + +let hashMap_t_same_bindings (#t : Type0) (hm hm' : hashMap_t_nes t) : Type0 = + forall (k:key). hashMap_t_find_s hm k == hashMap_t_find_s hm' k + +/// The final lemma about [try_resize] +val hashMap_try_resize_lem (#t : Type0) (hm : hashMap_t t) : + Lemma + (requires ( + hashMap_t_base_inv hm /\ + // However, the "full" invariant is broken, as we call [try_resize] + // only if the current number of entries is > the max load. + // + // There are two situations: + // - either we just reached the max load + // - or we were already saturated and can't resize + (let (dividend, divisor) = hm.max_load_factor in + hm.num_entries == hm.max_load + 1 \/ + length hm.slots * 2 * dividend > usize_max))) + (ensures ( + match hashMap_try_resize t hm with + | Fail _ -> False + | Return hm' -> + // The full invariant is now satisfied (the full invariant is "base + // invariant" + the map is not overloaded (or can't be resized because + // already too big) + hashMap_t_inv hm' /\ + // The length is the same + hashMap_t_len_s hm' = hashMap_t_len_s hm /\ + // It contains the same bindings as the initial map + hashMap_t_same_bindings hm' hm)) + +let hashMap_try_resize_lem #t hm = + hashMap_try_resize_lem_refin t hm; + hashMap_try_resize_s_simpl_lem hm + +(*** insert *) + +/// The high-level model (very close to the original function: we don't need something +/// very high level, just to clean it a bit) +let hashMap_insert_s + (#t : Type0) (self : hashMap_t t) (key : usize) (value : t) : + result (hashMap_t t) = + match hashMap_insert_no_resize t self key value with + | Fail e -> Fail e + | Return hm' -> + if hashMap_t_len_s hm' > hm'.max_load then + hashMap_try_resize t hm' + else Return hm' + +val hashMap_insert_lem_refin + (t : Type0) (self : hashMap_t t) (key : usize) (value : t) : + Lemma (requires True) + (ensures ( + match hashMap_insert t self key value, + hashMap_insert_s self key value + with + | Fail _, Fail _ -> True + | Return hm1, Return hm2 -> hm1 == hm2 + | _ -> False)) + +let hashMap_insert_lem_refin t self key value = () + +/// Helper +let hashMap_insert_bindings_lem + (t : Type0) (self : hashMap_t_nes t) (key : usize) (value : t) + (hm' hm'' : hashMap_t_nes t) : + Lemma + (requires ( + hashMap_s_updated_binding (hashMap_t_v self) key + (Some value) (hashMap_t_v hm') /\ + hashMap_t_same_bindings hm' hm'')) + (ensures ( + hashMap_s_updated_binding (hashMap_t_v self) key + (Some value) (hashMap_t_v hm''))) + = () + +val hashMap_insert_lem_aux + (#t : Type0) (self : hashMap_t t) (key : usize) (value : t) : + Lemma (requires (hashMap_t_inv self)) + (ensures ( + match hashMap_insert t self key value with + | Fail _ -> + // We can fail only if: + // - the key is not in the map and we need to add it + // - we are already saturated + hashMap_t_len_s self = usize_max /\ + None? (hashMap_t_find_s self key) + | Return hm' -> + // The invariant is preserved + hashMap_t_inv hm' /\ + // [key] maps to [value] and the other bindings are preserved + hashMap_s_updated_binding (hashMap_t_v self) key (Some value) (hashMap_t_v hm') /\ + // The length is incremented, iff we inserted a new key + (match hashMap_t_find_s self key with + | None -> hashMap_t_len_s hm' = hashMap_t_len_s self + 1 + | Some _ -> hashMap_t_len_s hm' = hashMap_t_len_s self))) + +#restart-solver +#push-options "--z3rlimit 200" +let hashMap_insert_lem_aux #t self key value = + hashMap_insert_no_resize_lem_s t self key value; + hashMap_insert_no_resize_s_lem (hashMap_t_v self) key value; + match hashMap_insert_no_resize t self key value with + | Fail _ -> () + | Return hm' -> + // Expanding the post of [hashMap_insert_no_resize_lem_s] + let self_v = hashMap_t_v self in + let hm'_v = Return?.v (hashMap_insert_no_resize_s self_v key value) in + assert(hashMap_t_base_inv hm'); + assert(hashMap_t_same_params hm' self); + assert(hashMap_t_v hm' == hm'_v); + assert(hashMap_s_len hm'_v == hashMap_t_len_s hm'); + // Expanding the post of [hashMap_insert_no_resize_s_lem] + assert(insert_post self_v key value hm'_v); + // Expanding [insert_post] + assert(hashMap_s_inv hm'_v); + assert( + match hashMap_s_find self_v key with + | None -> hashMap_s_len hm'_v = hashMap_s_len self_v + 1 + | Some _ -> hashMap_s_len hm'_v = hashMap_s_len self_v); + if hashMap_t_len_s hm' > hm'.max_load then + begin + hashMap_try_resize_lem hm'; + // Expanding the post of [hashMap_try_resize_lem] + let hm'' = Return?.v (hashMap_try_resize t hm') in + assert(hashMap_t_inv hm''); + let hm''_v = hashMap_t_v hm'' in + assert(forall k. hashMap_t_find_s hm'' k == hashMap_t_find_s hm' k); + assert(hashMap_t_len_s hm'' = hashMap_t_len_s hm'); // TODO + // Proving the post + assert(hashMap_t_inv hm''); + hashMap_insert_bindings_lem t self key value hm' hm''; + assert( + match hashMap_t_find_s self key with + | None -> hashMap_t_len_s hm'' = hashMap_t_len_s self + 1 + | Some _ -> hashMap_t_len_s hm'' = hashMap_t_len_s self) + end + else () +#pop-options + +let hashMap_insert_lem #t self key value = + hashMap_insert_lem_aux #t self key value + +(*** contains_key *) + +(**** contains_key_in_list *) + +val hashMap_contains_key_in_list_lem + (#t : Type0) (key : usize) (ls : list_t t) : + Lemma + (ensures ( + match hashMap_contains_key_in_list t key ls with + | Fail _ -> False + | Return b -> + b = Some? (slot_t_find_s key ls))) + + +#push-options "--fuel 1" +let rec hashMap_contains_key_in_list_lem #t key ls = + match ls with + | List_Cons ckey x ls0 -> + let b = ckey = key in + if b + then () + else + begin + hashMap_contains_key_in_list_lem key ls0; + match hashMap_contains_key_in_list t key ls0 with + | Fail _ -> () + | Return b0 -> () + end + | List_Nil -> () +#pop-options + +(**** contains_key *) + +val hashMap_contains_key_lem_aux + (#t : Type0) (self : hashMap_t_nes t) (key : usize) : + Lemma + (ensures ( + match hashMap_contains_key t self key with + | Fail _ -> False + | Return b -> b = Some? (hashMap_t_find_s self key))) + +let hashMap_contains_key_lem_aux #t self key = + begin match hash_key key with + | Fail _ -> () + | Return i -> + let v = self.slots in + let i0 = alloc_vec_Vec_len (list_t t) v in + begin match usize_rem i i0 with + | Fail _ -> () + | Return hash_mod -> + begin match alloc_vec_Vec_index_usize v hash_mod with + | Fail _ -> () + | Return l -> + hashMap_contains_key_in_list_lem key l; + begin match hashMap_contains_key_in_list t key l with + | Fail _ -> () + | Return b -> () + end + end + end + end + +/// The lemma in the .fsti +let hashMap_contains_key_lem #t self key = + hashMap_contains_key_lem_aux #t self key + +(*** get *) + +(**** get_in_list *) + +val hashMap_get_in_list_lem + (#t : Type0) (key : usize) (ls : list_t t) : + Lemma + (ensures ( + match hashMap_get_in_list t key ls, slot_t_find_s key ls with + | Fail _, None -> True + | Return x, Some x' -> x == x' + | _ -> False)) + +#push-options "--fuel 1" +let rec hashMap_get_in_list_lem #t key ls = + begin match ls with + | List_Cons ckey cvalue ls0 -> + let b = ckey = key in + if b + then () + else + begin + hashMap_get_in_list_lem key ls0; + match hashMap_get_in_list t key ls0 with + | Fail _ -> () + | Return x -> () + end + | List_Nil -> () + end +#pop-options + +(**** get *) + +val hashMap_get_lem_aux + (#t : Type0) (self : hashMap_t_nes t) (key : usize) : + Lemma + (ensures ( + match hashMap_get t self key, hashMap_t_find_s self key with + | Fail _, None -> True + | Return x, Some x' -> x == x' + | _ -> False)) + +let hashMap_get_lem_aux #t self key = + begin match hash_key key with + | Fail _ -> () + | Return i -> + let v = self.slots in + let i0 = alloc_vec_Vec_len (list_t t) v in + begin match usize_rem i i0 with + | Fail _ -> () + | Return hash_mod -> + begin match alloc_vec_Vec_index_usize v hash_mod with + | Fail _ -> () + | Return l -> + begin + hashMap_get_in_list_lem key l; + match hashMap_get_in_list t key l with + | Fail _ -> () + | Return x -> () + end + end + end + end + +/// .fsti +let hashMap_get_lem #t self key = hashMap_get_lem_aux #t self key + +(*** get_mut'fwd *) + + +(**** get_mut_in_list'fwd *) + +val hashMap_get_mut_in_list_loop_lem + (#t : Type0) (ls : list_t t) (key : usize) : + Lemma + (ensures ( + match hashMap_get_mut_in_list_loop t ls key, slot_t_find_s key ls with + | Fail _, None -> True + | Return x, Some x' -> x == x' + | _ -> False)) + +#push-options "--fuel 1" +let rec hashMap_get_mut_in_list_loop_lem #t ls key = + begin match ls with + | List_Cons ckey cvalue ls0 -> + let b = ckey = key in + if b + then () + else + begin + hashMap_get_mut_in_list_loop_lem ls0 key; + match hashMap_get_mut_in_list_loop t ls0 key with + | Fail _ -> () + | Return x -> () + end + | List_Nil -> () + end +#pop-options + +(**** get_mut'fwd *) + +val hashMap_get_mut_lem_aux + (#t : Type0) (self : hashMap_t_nes t) (key : usize) : + Lemma + (ensures ( + match hashMap_get_mut t self key, hashMap_t_find_s self key with + | Fail _, None -> True + | Return x, Some x' -> x == x' + | _ -> False)) + +let hashMap_get_mut_lem_aux #t self key = + begin match hash_key key with + | Fail _ -> () + | Return i -> + let v = self.slots in + let i0 = alloc_vec_Vec_len (list_t t) v in + begin match usize_rem i i0 with + | Fail _ -> () + | Return hash_mod -> + begin match alloc_vec_Vec_index_usize v hash_mod with + | Fail _ -> () + | Return l -> + begin + hashMap_get_mut_in_list_loop_lem l key; + match hashMap_get_mut_in_list_loop t l key with + | Fail _ -> () + | Return x -> () + end + end + end + end + +let hashMap_get_mut_lem #t self key = + hashMap_get_mut_lem_aux #t self key + +(*** get_mut'back *) + +(**** get_mut_in_list'back *) + +val hashMap_get_mut_in_list_loop_back_lem + (#t : Type0) (ls : list_t t) (key : usize) (ret : t) : + Lemma + (requires (Some? (slot_t_find_s key ls))) + (ensures ( + match hashMap_get_mut_in_list_loop_back t ls key ret with + | Fail _ -> False + | Return ls' -> list_t_v ls' == find_update (same_key key) (list_t_v ls) (key,ret) + | _ -> False)) + +#push-options "--fuel 1" +let rec hashMap_get_mut_in_list_loop_back_lem #t ls key ret = + begin match ls with + | List_Cons ckey cvalue ls0 -> + let b = ckey = key in + if b + then let ls1 = List_Cons ckey ret ls0 in () + else + begin + hashMap_get_mut_in_list_loop_back_lem ls0 key ret; + match hashMap_get_mut_in_list_loop_back t ls0 key ret with + | Fail _ -> () + | Return l -> let ls1 = List_Cons ckey cvalue l in () + end + | List_Nil -> () + end +#pop-options + +(**** get_mut'back *) + +/// Refinement lemma +val hashMap_get_mut_back_lem_refin + (#t : Type0) (self : hashMap_t t{length self.slots > 0}) + (key : usize) (ret : t) : + Lemma + (requires (Some? (hashMap_t_find_s self key))) + (ensures ( + match hashMap_get_mut_back t self key ret with + | Fail _ -> False + | Return hm' -> + hashMap_t_v hm' == hashMap_insert_no_fail_s (hashMap_t_v self) key ret)) + +let hashMap_get_mut_back_lem_refin #t self key ret = + begin match hash_key key with + | Fail _ -> () + | Return i -> + let i0 = self.num_entries in + let p = self.max_load_factor in + let i1 = self.max_load in + let v = self.slots in + let i2 = alloc_vec_Vec_len (list_t t) v in + begin match usize_rem i i2 with + | Fail _ -> () + | Return hash_mod -> + begin match alloc_vec_Vec_index_usize v hash_mod with + | Fail _ -> () + | Return l -> + begin + hashMap_get_mut_in_list_loop_back_lem l key ret; + match hashMap_get_mut_in_list_loop_back t l key ret with + | Fail _ -> () + | Return l0 -> + begin match alloc_vec_Vec_update_usize v hash_mod l0 with + | Fail _ -> () + | Return v0 -> let self0 = MkhashMap_t i0 p i1 v0 in () + end + end + end + end + end + +/// Final lemma +val hashMap_get_mut_back_lem_aux + (#t : Type0) (hm : hashMap_t t) + (key : usize) (ret : t) : + Lemma + (requires ( + hashMap_t_inv hm /\ + Some? (hashMap_t_find_s hm key))) + (ensures ( + match hashMap_get_mut_back t hm key ret with + | Fail _ -> False + | Return hm' -> + // Functional spec + hashMap_t_v hm' == hashMap_insert_no_fail_s (hashMap_t_v hm) key ret /\ + // The invariant is preserved + hashMap_t_inv hm' /\ + // The length is preserved + hashMap_t_len_s hm' = hashMap_t_len_s hm /\ + // [key] maps to [value] + hashMap_t_find_s hm' key == Some ret /\ + // The other bindings are preserved + (forall k'. k' <> key ==> hashMap_t_find_s hm' k' == hashMap_t_find_s hm k'))) + +let hashMap_get_mut_back_lem_aux #t hm key ret = + let hm_v = hashMap_t_v hm in + hashMap_get_mut_back_lem_refin hm key ret; + match hashMap_get_mut_back t hm key ret with + | Fail _ -> assert(False) + | Return hm' -> + hashMap_insert_no_fail_s_lem hm_v key ret + +/// .fsti +let hashMap_get_mut_back_lem #t hm key ret = hashMap_get_mut_back_lem_aux hm key ret + +(*** remove'fwd *) + +val hashMap_remove_from_list_lem + (#t : Type0) (key : usize) (ls : list_t t) : + Lemma + (ensures ( + match hashMap_remove_from_list t key ls with + | Fail _ -> False + | Return opt_x -> + opt_x == slot_t_find_s key ls /\ + (Some? opt_x ==> length (slot_t_v ls) > 0))) + +#push-options "--fuel 1" +let rec hashMap_remove_from_list_lem #t key ls = + begin match ls with + | List_Cons ckey x tl -> + let b = ckey = key in + if b + then + let mv_ls = core_mem_replace (list_t t) (List_Cons ckey x tl) List_Nil in + begin match mv_ls with + | List_Cons i cvalue tl0 -> () + | List_Nil -> () + end + else + begin + hashMap_remove_from_list_lem key tl; + match hashMap_remove_from_list t key tl with + | Fail _ -> () + | Return opt -> () + end + | List_Nil -> () + end +#pop-options + +val hashMap_remove_lem_aux + (#t : Type0) (self : hashMap_t t) (key : usize) : + Lemma + (requires ( + // We need the invariant to prove that upon decrementing the entries counter, + // the counter doesn't become negative + hashMap_t_inv self)) + (ensures ( + match hashMap_remove t self key with + | Fail _ -> False + | Return opt_x -> opt_x == hashMap_t_find_s self key)) + +let hashMap_remove_lem_aux #t self key = + begin match hash_key key with + | Fail _ -> () + | Return i -> + let i0 = self.num_entries in + let v = self.slots in + let i1 = alloc_vec_Vec_len (list_t t) v in + begin match usize_rem i i1 with + | Fail _ -> () + | Return hash_mod -> + begin match alloc_vec_Vec_index_usize v hash_mod with + | Fail _ -> () + | Return l -> + begin + hashMap_remove_from_list_lem key l; + match hashMap_remove_from_list t key l with + | Fail _ -> () + | Return x -> + begin match x with + | None -> () + | Some x0 -> + begin + assert(l == index v hash_mod); + assert(length (list_t_v #t l) > 0); + length_flatten_index (hashMap_t_v self) hash_mod; + match usize_sub i0 1 with + | Fail _ -> () + | Return _ -> () + end + end + end + end + end + end + +/// .fsti +let hashMap_remove_lem #t self key = hashMap_remove_lem_aux #t self key + +(*** remove'back *) + +(**** Refinement proofs *) + +/// High-level model for [remove_from_list'back] +let hashMap_remove_from_list_s + (#t : Type0) (key : usize) (ls : slot_s t) : + slot_s t = + filter_one (not_same_key key) ls + +/// Refinement lemma +val hashMap_remove_from_list_back_lem_refin + (#t : Type0) (key : usize) (ls : list_t t) : + Lemma + (ensures ( + match hashMap_remove_from_list_back t key ls with + | Fail _ -> False + | Return ls' -> + list_t_v ls' == hashMap_remove_from_list_s key (list_t_v ls) /\ + // The length is decremented, iff the key was in the slot + (let len = length (list_t_v ls) in + let len' = length (list_t_v ls') in + match slot_s_find key (list_t_v ls) with + | None -> len = len' + | Some _ -> len = len' + 1))) + +#push-options "--fuel 1" +let rec hashMap_remove_from_list_back_lem_refin #t key ls = + begin match ls with + | List_Cons ckey x tl -> + let b = ckey = key in + if b + then + let mv_ls = core_mem_replace (list_t t) (List_Cons ckey x tl) List_Nil in + begin match mv_ls with + | List_Cons i cvalue tl0 -> () + | List_Nil -> () + end + else + begin + hashMap_remove_from_list_back_lem_refin key tl; + match hashMap_remove_from_list_back t key tl with + | Fail _ -> () + | Return l -> let ls0 = List_Cons ckey x l in () + end + | List_Nil -> () + end +#pop-options + +/// High-level model for [remove_from_list'back] +let hashMap_remove_s + (#t : Type0) (self : hashMap_s_nes t) (key : usize) : + hashMap_s t = + let len = length self in + let hash = hash_mod_key key len in + let slot = index self hash in + let slot' = hashMap_remove_from_list_s key slot in + list_update self hash slot' + +/// Refinement lemma +val hashMap_remove_back_lem_refin + (#t : Type0) (self : hashMap_t_nes t) (key : usize) : + Lemma + (requires ( + // We need the invariant to prove that upon decrementing the entries counter, + // the counter doesn't become negative + hashMap_t_inv self)) + (ensures ( + match hashMap_remove_back t self key with + | Fail _ -> False + | Return hm' -> + hashMap_t_same_params hm' self /\ + hashMap_t_v hm' == hashMap_remove_s (hashMap_t_v self) key /\ + // The length is decremented iff the key was in the map + (let len = hashMap_t_len_s self in + let len' = hashMap_t_len_s hm' in + match hashMap_t_find_s self key with + | None -> len = len' + | Some _ -> len = len' + 1))) + +let hashMap_remove_back_lem_refin #t self key = + begin match hash_key key with + | Fail _ -> () + | Return i -> + let i0 = self.num_entries in + let p = self.max_load_factor in + let i1 = self.max_load in + let v = self.slots in + let i2 = alloc_vec_Vec_len (list_t t) v in + begin match usize_rem i i2 with + | Fail _ -> () + | Return hash_mod -> + begin match alloc_vec_Vec_index_usize v hash_mod with + | Fail _ -> () + | Return l -> + begin + hashMap_remove_from_list_lem key l; + match hashMap_remove_from_list t key l with + | Fail _ -> () + | Return x -> + begin match x with + | None -> + begin + hashMap_remove_from_list_back_lem_refin key l; + match hashMap_remove_from_list_back t key l with + | Fail _ -> () + | Return l0 -> + begin + length_flatten_update (slots_t_v v) hash_mod (list_t_v l0); + match alloc_vec_Vec_update_usize v hash_mod l0 with + | Fail _ -> () + | Return v0 -> () + end + end + | Some x0 -> + begin + assert(l == index v hash_mod); + assert(length (list_t_v #t l) > 0); + length_flatten_index (hashMap_t_v self) hash_mod; + match usize_sub i0 1 with + | Fail _ -> () + | Return i3 -> + begin + hashMap_remove_from_list_back_lem_refin key l; + match hashMap_remove_from_list_back t key l with + | Fail _ -> () + | Return l0 -> + begin + length_flatten_update (slots_t_v v) hash_mod (list_t_v l0); + match alloc_vec_Vec_update_usize v hash_mod l0 with + | Fail _ -> () + | Return v0 -> () + end + end + end + end + end + end + end + end + +(**** Invariants, high-level properties *) + +val hashMap_remove_from_list_s_lem + (#t : Type0) (k : usize) (slot : slot_s t) (len : usize{len > 0}) (i : usize) : + Lemma + (requires (slot_s_inv len i slot)) + (ensures ( + let slot' = hashMap_remove_from_list_s k slot in + slot_s_inv len i slot' /\ + slot_s_find k slot' == None /\ + (forall (k':key{k' <> k}). slot_s_find k' slot' == slot_s_find k' slot) /\ + // This postcondition is necessary to prove that the invariant is preserved + // in the recursive calls. This allows us to do the proof in one go. + (forall (b:binding t). for_all (binding_neq b) slot ==> for_all (binding_neq b) slot') + )) + +#push-options "--fuel 1" +let rec hashMap_remove_from_list_s_lem #t key slot len i = + match slot with + | [] -> () + | (k',v) :: slot' -> + if k' <> key then + begin + hashMap_remove_from_list_s_lem key slot' len i; + let slot'' = hashMap_remove_from_list_s key slot' in + assert(for_all (same_hash_mod_key len i) ((k',v)::slot'')); + assert(for_all (binding_neq (k',v)) slot'); // Triggers instanciation + assert(for_all (binding_neq (k',v)) slot'') + end + else + begin + assert(for_all (binding_neq (k',v)) slot'); + for_all_binding_neq_find_lem key v slot' + end +#pop-options + +val hashMap_remove_s_lem + (#t : Type0) (self : hashMap_s_nes t) (key : usize) : + Lemma + (requires (hashMap_s_inv self)) + (ensures ( + let hm' = hashMap_remove_s self key in + // The invariant is preserved + hashMap_s_inv hm' /\ + // We updated the binding + hashMap_s_updated_binding self key None hm')) + +let hashMap_remove_s_lem #t self key = + let len = length self in + let hash = hash_mod_key key len in + let slot = index self hash in + hashMap_remove_from_list_s_lem key slot len hash; + let slot' = hashMap_remove_from_list_s key slot in + let hm' = list_update self hash slot' in + assert(hashMap_s_inv self) + +/// Final lemma about [remove'back] +val hashMap_remove_back_lem_aux + (#t : Type0) (self : hashMap_t t) (key : usize) : + Lemma + (requires (hashMap_t_inv self)) + (ensures ( + match hashMap_remove_back t self key with + | Fail _ -> False + | Return hm' -> + hashMap_t_inv self /\ + hashMap_t_same_params hm' self /\ + // We updated the binding + hashMap_s_updated_binding (hashMap_t_v self) key None (hashMap_t_v hm') /\ + hashMap_t_v hm' == hashMap_remove_s (hashMap_t_v self) key /\ + // The length is decremented iff the key was in the map + (let len = hashMap_t_len_s self in + let len' = hashMap_t_len_s hm' in + match hashMap_t_find_s self key with + | None -> len = len' + | Some _ -> len = len' + 1))) + +let hashMap_remove_back_lem_aux #t self key = + hashMap_remove_back_lem_refin self key; + hashMap_remove_s_lem (hashMap_t_v self) key + +/// .fsti +let hashMap_remove_back_lem #t self key = + hashMap_remove_back_lem_aux #t self key |