| ------------------------------------------------------------------------------ |
| -- -- |
| -- GNAT LIBRARY COMPONENTS -- |
| -- -- |
| -- A D A . C O N T A I N E R S . F O R M A L _ O R D E R E D _ S E T S -- |
| -- -- |
| -- S p e c -- |
| -- -- |
| -- Copyright (C) 2004-2022, Free Software Foundation, Inc. -- |
| -- -- |
| -- This specification is derived from the Ada Reference Manual for use with -- |
| -- GNAT. The copyright notice above, and the license provisions that follow -- |
| -- apply solely to the contents of the part following the private keyword. -- |
| -- -- |
| -- GNAT is free software; you can redistribute it and/or modify it under -- |
| -- terms of the GNU General Public License as published by the Free Soft- -- |
| -- ware Foundation; either version 3, or (at your option) any later ver- -- |
| -- sion. GNAT is distributed in the hope that it will be useful, but WITH- -- |
| -- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY -- |
| -- or FITNESS FOR A PARTICULAR PURPOSE. -- |
| -- -- |
| -- As a special exception under Section 7 of GPL version 3, you are granted -- |
| -- additional permissions described in the GCC Runtime Library Exception, -- |
| -- version 3.1, as published by the Free Software Foundation. -- |
| -- -- |
| -- You should have received a copy of the GNU General Public License and -- |
| -- a copy of the GCC Runtime Library Exception along with this program; -- |
| -- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see -- |
| -- <http://www.gnu.org/licenses/>. -- |
| ------------------------------------------------------------------------------ |
| |
| -- This spec is derived from package Ada.Containers.Bounded_Ordered_Sets in |
| -- the Ada 2012 RM. The modifications are meant to facilitate formal proofs by |
| -- making it easier to express properties, and by making the specification of |
| -- this unit compatible with SPARK 2014. Note that the API of this unit may be |
| -- subject to incompatible changes as SPARK 2014 evolves. |
| |
| -- The modifications are: |
| |
| -- A parameter for the container is added to every function reading the |
| -- content of a container: Key, Element, Next, Query_Element, Previous, |
| -- Has_Element, Iterate, Reverse_Iterate. This change is motivated by the |
| -- need to have cursors which are valid on different containers (typically |
| -- a container C and its previous version C'Old) for expressing properties, |
| -- which is not possible if cursors encapsulate an access to the underlying |
| -- container. The operators "<" and ">" that could not be modified that way |
| -- have been removed. |
| |
| with Ada.Containers.Functional_Maps; |
| with Ada.Containers.Functional_Sets; |
| with Ada.Containers.Functional_Vectors; |
| private with Ada.Containers.Red_Black_Trees; |
| |
| generic |
| type Element_Type is private; |
| |
| with function "<" (Left, Right : Element_Type) return Boolean is <>; |
| |
| package Ada.Containers.Formal_Ordered_Sets with |
| SPARK_Mode |
| is |
| -- Contracts in this unit are meant for analysis only, not for run-time |
| -- checking. |
| |
| pragma Assertion_Policy (Pre => Ignore); |
| pragma Assertion_Policy (Post => Ignore); |
| pragma Assertion_Policy (Contract_Cases => Ignore); |
| pragma Annotate (CodePeer, Skip_Analysis); |
| |
| function Equivalent_Elements (Left, Right : Element_Type) return Boolean |
| with |
| Global => null, |
| Post => |
| Equivalent_Elements'Result = |
| (not (Left < Right) and not (Right < Left)); |
| pragma Annotate (GNATprove, Inline_For_Proof, Equivalent_Elements); |
| |
| type Set (Capacity : Count_Type) is private with |
| Iterable => (First => First, |
| Next => Next, |
| Has_Element => Has_Element, |
| Element => Element), |
| Default_Initial_Condition => Is_Empty (Set); |
| pragma Preelaborable_Initialization (Set); |
| |
| type Cursor is record |
| Node : Count_Type; |
| end record; |
| |
| No_Element : constant Cursor := (Node => 0); |
| |
| function Length (Container : Set) return Count_Type with |
| Global => null, |
| Post => Length'Result <= Container.Capacity; |
| |
| pragma Unevaluated_Use_Of_Old (Allow); |
| |
| package Formal_Model with Ghost is |
| subtype Positive_Count_Type is Count_Type range 1 .. Count_Type'Last; |
| |
| package M is new Ada.Containers.Functional_Sets |
| (Element_Type => Element_Type, |
| Equivalent_Elements => Equivalent_Elements); |
| |
| function "=" |
| (Left : M.Set; |
| Right : M.Set) return Boolean renames M."="; |
| |
| function "<=" |
| (Left : M.Set; |
| Right : M.Set) return Boolean renames M."<="; |
| |
| package E is new Ada.Containers.Functional_Vectors |
| (Element_Type => Element_Type, |
| Index_Type => Positive_Count_Type); |
| |
| function "=" |
| (Left : E.Sequence; |
| Right : E.Sequence) return Boolean renames E."="; |
| |
| function "<" |
| (Left : E.Sequence; |
| Right : E.Sequence) return Boolean renames E."<"; |
| |
| function "<=" |
| (Left : E.Sequence; |
| Right : E.Sequence) return Boolean renames E."<="; |
| |
| function E_Bigger_Than_Range |
| (Container : E.Sequence; |
| Fst : Positive_Count_Type; |
| Lst : Count_Type; |
| Item : Element_Type) return Boolean |
| with |
| Global => null, |
| Pre => Lst <= E.Length (Container), |
| Post => |
| E_Bigger_Than_Range'Result = |
| (for all I in Fst .. Lst => E.Get (Container, I) < Item); |
| pragma Annotate (GNATprove, Inline_For_Proof, E_Bigger_Than_Range); |
| |
| function E_Smaller_Than_Range |
| (Container : E.Sequence; |
| Fst : Positive_Count_Type; |
| Lst : Count_Type; |
| Item : Element_Type) return Boolean |
| with |
| Global => null, |
| Pre => Lst <= E.Length (Container), |
| Post => |
| E_Smaller_Than_Range'Result = |
| (for all I in Fst .. Lst => Item < E.Get (Container, I)); |
| pragma Annotate (GNATprove, Inline_For_Proof, E_Smaller_Than_Range); |
| |
| function E_Is_Find |
| (Container : E.Sequence; |
| Item : Element_Type; |
| Position : Count_Type) return Boolean |
| with |
| Global => null, |
| Pre => Position - 1 <= E.Length (Container), |
| Post => |
| E_Is_Find'Result = |
| |
| ((if Position > 0 then |
| E_Bigger_Than_Range (Container, 1, Position - 1, Item)) |
| |
| and (if Position < E.Length (Container) then |
| E_Smaller_Than_Range |
| (Container, |
| Position + 1, |
| E.Length (Container), |
| Item))); |
| pragma Annotate (GNATprove, Inline_For_Proof, E_Is_Find); |
| |
| function Find |
| (Container : E.Sequence; |
| Item : Element_Type) return Count_Type |
| -- Search for Item in Container |
| |
| with |
| Global => null, |
| Post => |
| (if Find'Result > 0 then |
| Find'Result <= E.Length (Container) |
| and Equivalent_Elements (Item, E.Get (Container, Find'Result))); |
| |
| function E_Elements_Included |
| (Left : E.Sequence; |
| Right : E.Sequence) return Boolean |
| -- The elements of Left are contained in Right |
| |
| with |
| Global => null, |
| Post => |
| E_Elements_Included'Result = |
| (for all I in 1 .. E.Length (Left) => |
| Find (Right, E.Get (Left, I)) > 0 |
| and then E.Get (Right, Find (Right, E.Get (Left, I))) = |
| E.Get (Left, I)); |
| pragma Annotate (GNATprove, Inline_For_Proof, E_Elements_Included); |
| |
| function E_Elements_Included |
| (Left : E.Sequence; |
| Model : M.Set; |
| Right : E.Sequence) return Boolean |
| -- The elements of Container contained in Model are in Right |
| |
| with |
| Global => null, |
| Post => |
| E_Elements_Included'Result = |
| (for all I in 1 .. E.Length (Left) => |
| (if M.Contains (Model, E.Get (Left, I)) then |
| Find (Right, E.Get (Left, I)) > 0 |
| and then E.Get (Right, Find (Right, E.Get (Left, I))) = |
| E.Get (Left, I))); |
| pragma Annotate (GNATprove, Inline_For_Proof, E_Elements_Included); |
| |
| function E_Elements_Included |
| (Container : E.Sequence; |
| Model : M.Set; |
| Left : E.Sequence; |
| Right : E.Sequence) return Boolean |
| -- The elements of Container contained in Model are in Left and others |
| -- are in Right. |
| |
| with |
| Global => null, |
| Post => |
| E_Elements_Included'Result = |
| (for all I in 1 .. E.Length (Container) => |
| (if M.Contains (Model, E.Get (Container, I)) then |
| Find (Left, E.Get (Container, I)) > 0 |
| and then E.Get (Left, Find (Left, E.Get (Container, I))) = |
| E.Get (Container, I) |
| else |
| Find (Right, E.Get (Container, I)) > 0 |
| and then E.Get (Right, Find (Right, E.Get (Container, I))) = |
| E.Get (Container, I))); |
| pragma Annotate (GNATprove, Inline_For_Proof, E_Elements_Included); |
| |
| package P is new Ada.Containers.Functional_Maps |
| (Key_Type => Cursor, |
| Element_Type => Positive_Count_Type, |
| Equivalent_Keys => "=", |
| Enable_Handling_Of_Equivalence => False); |
| |
| function "=" |
| (Left : P.Map; |
| Right : P.Map) return Boolean renames P."="; |
| |
| function "<=" |
| (Left : P.Map; |
| Right : P.Map) return Boolean renames P."<="; |
| |
| function P_Positions_Shifted |
| (Small : P.Map; |
| Big : P.Map; |
| Cut : Positive_Count_Type; |
| Count : Count_Type := 1) return Boolean |
| with |
| Global => null, |
| Post => |
| P_Positions_Shifted'Result = |
| |
| -- Big contains all cursors of Small |
| |
| (P.Keys_Included (Small, Big) |
| |
| -- Cursors located before Cut are not moved, cursors located |
| -- after are shifted by Count. |
| |
| and (for all I of Small => |
| (if P.Get (Small, I) < Cut then |
| P.Get (Big, I) = P.Get (Small, I) |
| else |
| P.Get (Big, I) - Count = P.Get (Small, I))) |
| |
| -- New cursors of Big (if any) are between Cut and Cut - 1 + |
| -- Count. |
| |
| and (for all I of Big => |
| P.Has_Key (Small, I) |
| or P.Get (Big, I) - Count in Cut - Count .. Cut - 1)); |
| |
| function Mapping_Preserved |
| (E_Left : E.Sequence; |
| E_Right : E.Sequence; |
| P_Left : P.Map; |
| P_Right : P.Map) return Boolean |
| with |
| Ghost, |
| Global => null, |
| Post => |
| (if Mapping_Preserved'Result then |
| |
| -- Right contains all the cursors of Left |
| |
| P.Keys_Included (P_Left, P_Right) |
| |
| -- Right contains all the elements of Left |
| |
| and E_Elements_Included (E_Left, E_Right) |
| |
| -- Mappings from cursors to elements induced by E_Left, P_Left |
| -- and E_Right, P_Right are the same. |
| |
| and (for all C of P_Left => |
| E.Get (E_Left, P.Get (P_Left, C)) = |
| E.Get (E_Right, P.Get (P_Right, C)))); |
| |
| function Mapping_Preserved_Except |
| (E_Left : E.Sequence; |
| E_Right : E.Sequence; |
| P_Left : P.Map; |
| P_Right : P.Map; |
| Position : Cursor) return Boolean |
| with |
| Ghost, |
| Global => null, |
| Post => |
| (if Mapping_Preserved_Except'Result then |
| |
| -- Right contains all the cursors of Left |
| |
| P.Keys_Included (P_Left, P_Right) |
| |
| -- Mappings from cursors to elements induced by E_Left, P_Left |
| -- and E_Right, P_Right are the same except for Position. |
| |
| and (for all C of P_Left => |
| (if C /= Position then |
| E.Get (E_Left, P.Get (P_Left, C)) = |
| E.Get (E_Right, P.Get (P_Right, C))))); |
| |
| function Model (Container : Set) return M.Set with |
| -- The high-level model of a set is a set of elements. Neither cursors |
| -- nor order of elements are represented in this model. Elements are |
| -- modeled up to equivalence. |
| |
| Ghost, |
| Global => null, |
| Post => M.Length (Model'Result) = Length (Container); |
| |
| function Elements (Container : Set) return E.Sequence with |
| -- The Elements sequence represents the underlying list structure of |
| -- sets that is used for iteration. It stores the actual values of |
| -- elements in the set. It does not model cursors. |
| |
| Ghost, |
| Global => null, |
| Post => |
| E.Length (Elements'Result) = Length (Container) |
| |
| -- It only contains keys contained in Model |
| |
| and (for all Item of Elements'Result => |
| M.Contains (Model (Container), Item)) |
| |
| -- It contains all the elements contained in Model |
| |
| and (for all Item of Model (Container) => |
| (Find (Elements'Result, Item) > 0 |
| and then Equivalent_Elements |
| (E.Get (Elements'Result, Find (Elements'Result, Item)), |
| Item))) |
| |
| -- It is sorted in increasing order |
| |
| and (for all I in 1 .. Length (Container) => |
| Find (Elements'Result, E.Get (Elements'Result, I)) = I |
| and |
| E_Is_Find |
| (Elements'Result, E.Get (Elements'Result, I), I)); |
| pragma Annotate (GNATprove, Iterable_For_Proof, "Model", Elements); |
| |
| function Positions (Container : Set) return P.Map with |
| -- The Positions map is used to model cursors. It only contains valid |
| -- cursors and maps them to their position in the container. |
| |
| Ghost, |
| Global => null, |
| Post => |
| not P.Has_Key (Positions'Result, No_Element) |
| |
| -- Positions of cursors are smaller than the container's length |
| |
| and then |
| (for all I of Positions'Result => |
| P.Get (Positions'Result, I) in 1 .. Length (Container) |
| |
| -- No two cursors have the same position. Note that we do not |
| -- state that there is a cursor in the map for each position, as |
| -- it is rarely needed. |
| |
| and then |
| (for all J of Positions'Result => |
| (if P.Get (Positions'Result, I) = P.Get (Positions'Result, J) |
| then I = J))); |
| |
| procedure Lift_Abstraction_Level (Container : Set) with |
| -- Lift_Abstraction_Level is a ghost procedure that does nothing but |
| -- assume that we can access the same elements by iterating over |
| -- positions or cursors. |
| -- This information is not generally useful except when switching from |
| -- a low-level, cursor-aware view of a container, to a high-level, |
| -- position-based view. |
| |
| Ghost, |
| Global => null, |
| Post => |
| (for all Item of Elements (Container) => |
| (for some I of Positions (Container) => |
| E.Get (Elements (Container), P.Get (Positions (Container), I)) = |
| Item)); |
| |
| function Contains |
| (C : M.Set; |
| K : Element_Type) return Boolean renames M.Contains; |
| -- To improve readability of contracts, we rename the function used to |
| -- search for an element in the model to Contains. |
| |
| end Formal_Model; |
| use Formal_Model; |
| |
| Empty_Set : constant Set; |
| |
| function "=" (Left, Right : Set) return Boolean with |
| Global => null, |
| Post => |
| |
| -- If two sets are equal, they contain the same elements in the same |
| -- order. |
| |
| (if "="'Result then Elements (Left) = Elements (Right) |
| |
| -- If they are different, then they do not contain the same elements |
| |
| else |
| not E_Elements_Included (Elements (Left), Elements (Right)) |
| or not E_Elements_Included (Elements (Right), Elements (Left))); |
| |
| function Equivalent_Sets (Left, Right : Set) return Boolean with |
| Global => null, |
| Post => Equivalent_Sets'Result = (Model (Left) = Model (Right)); |
| |
| function To_Set (New_Item : Element_Type) return Set with |
| Global => null, |
| Post => |
| M.Is_Singleton (Model (To_Set'Result), New_Item) |
| and Length (To_Set'Result) = 1 |
| and E.Get (Elements (To_Set'Result), 1) = New_Item; |
| |
| function Is_Empty (Container : Set) return Boolean with |
| Global => null, |
| Post => Is_Empty'Result = (Length (Container) = 0); |
| |
| procedure Clear (Container : in out Set) with |
| Global => null, |
| Post => Length (Container) = 0 and M.Is_Empty (Model (Container)); |
| |
| procedure Assign (Target : in out Set; Source : Set) with |
| Global => null, |
| Pre => Target.Capacity >= Length (Source), |
| Post => |
| Model (Target) = Model (Source) |
| and Elements (Target) = Elements (Source) |
| and Length (Target) = Length (Source); |
| |
| function Copy (Source : Set; Capacity : Count_Type := 0) return Set with |
| Global => null, |
| Pre => Capacity = 0 or else Capacity >= Source.Capacity, |
| Post => |
| Model (Copy'Result) = Model (Source) |
| and Elements (Copy'Result) = Elements (Source) |
| and Positions (Copy'Result) = Positions (Source) |
| and (if Capacity = 0 then |
| Copy'Result.Capacity = Source.Capacity |
| else |
| Copy'Result.Capacity = Capacity); |
| |
| function Element |
| (Container : Set; |
| Position : Cursor) return Element_Type |
| with |
| Global => null, |
| Pre => Has_Element (Container, Position), |
| Post => |
| Element'Result = |
| E.Get (Elements (Container), P.Get (Positions (Container), Position)); |
| pragma Annotate (GNATprove, Inline_For_Proof, Element); |
| |
| procedure Replace_Element |
| (Container : in out Set; |
| Position : Cursor; |
| New_Item : Element_Type) |
| with |
| Global => null, |
| Pre => Has_Element (Container, Position), |
| Post => |
| Length (Container) = Length (Container)'Old |
| |
| -- Position now maps to New_Item |
| |
| and Element (Container, Position) = New_Item |
| |
| -- New_Item is contained in Container |
| |
| and Contains (Model (Container), New_Item) |
| |
| -- Other elements are preserved |
| |
| and M.Included_Except |
| (Model (Container)'Old, |
| Model (Container), |
| Element (Container, Position)'Old) |
| and M.Included_Except |
| (Model (Container), |
| Model (Container)'Old, |
| New_Item) |
| |
| -- Mapping from cursors to elements is preserved |
| |
| and Mapping_Preserved_Except |
| (E_Left => Elements (Container)'Old, |
| E_Right => Elements (Container), |
| P_Left => Positions (Container)'Old, |
| P_Right => Positions (Container), |
| Position => Position) |
| and Positions (Container) = Positions (Container)'Old; |
| |
| function Constant_Reference |
| (Container : aliased Set; |
| Position : Cursor) return not null access constant Element_Type |
| with |
| Global => null, |
| Pre => Has_Element (Container, Position), |
| Post => |
| Constant_Reference'Result.all = |
| E.Get (Elements (Container), P.Get (Positions (Container), Position)); |
| |
| procedure Move (Target : in out Set; Source : in out Set) with |
| Global => null, |
| Pre => Target.Capacity >= Length (Source), |
| Post => |
| Model (Target) = Model (Source)'Old |
| and Elements (Target) = Elements (Source)'Old |
| and Length (Source)'Old = Length (Target) |
| and Length (Source) = 0; |
| |
| procedure Insert |
| (Container : in out Set; |
| New_Item : Element_Type; |
| Position : out Cursor; |
| Inserted : out Boolean) |
| with |
| Global => null, |
| Pre => |
| Length (Container) < Container.Capacity |
| or Contains (Container, New_Item), |
| Post => |
| Contains (Container, New_Item) |
| and Has_Element (Container, Position) |
| and Equivalent_Elements (Element (Container, Position), New_Item) |
| and E_Is_Find |
| (Elements (Container), |
| New_Item, |
| P.Get (Positions (Container), Position)), |
| Contract_Cases => |
| |
| -- If New_Item is already in Container, it is not modified and Inserted |
| -- is set to False. |
| |
| (Contains (Container, New_Item) => |
| not Inserted |
| and Model (Container) = Model (Container)'Old |
| and Elements (Container) = Elements (Container)'Old |
| and Positions (Container) = Positions (Container)'Old, |
| |
| -- Otherwise, New_Item is inserted in Container and Inserted is set to |
| -- True |
| |
| others => |
| Inserted |
| and Length (Container) = Length (Container)'Old + 1 |
| |
| -- Position now maps to New_Item |
| |
| and Element (Container, Position) = New_Item |
| |
| -- Other elements are preserved |
| |
| and Model (Container)'Old <= Model (Container) |
| and M.Included_Except |
| (Model (Container), |
| Model (Container)'Old, |
| New_Item) |
| |
| -- The elements of Container located before Position are preserved |
| |
| and E.Range_Equal |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => 1, |
| Lst => P.Get (Positions (Container), Position) - 1) |
| |
| -- Other elements are shifted by 1 |
| |
| and E.Range_Shifted |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => P.Get (Positions (Container), Position), |
| Lst => Length (Container)'Old, |
| Offset => 1) |
| |
| -- A new cursor has been inserted at position Position in |
| -- Container. |
| |
| and P_Positions_Shifted |
| (Positions (Container)'Old, |
| Positions (Container), |
| Cut => P.Get (Positions (Container), Position))); |
| |
| procedure Insert |
| (Container : in out Set; |
| New_Item : Element_Type) |
| with |
| Global => null, |
| Pre => |
| Length (Container) < Container.Capacity |
| and then (not Contains (Container, New_Item)), |
| Post => |
| Length (Container) = Length (Container)'Old + 1 |
| and Contains (Container, New_Item) |
| |
| -- New_Item is inserted in the set |
| |
| and E.Get (Elements (Container), |
| Find (Elements (Container), New_Item)) = New_Item |
| |
| -- Other mappings are preserved |
| |
| and Model (Container)'Old <= Model (Container) |
| and M.Included_Except |
| (Model (Container), |
| Model (Container)'Old, |
| New_Item) |
| |
| -- The elements of Container located before New_Item are preserved |
| |
| and E.Range_Equal |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => 1, |
| Lst => Find (Elements (Container), New_Item) - 1) |
| |
| -- Other elements are shifted by 1 |
| |
| and E.Range_Shifted |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => Find (Elements (Container), New_Item), |
| Lst => Length (Container)'Old, |
| Offset => 1) |
| |
| -- A new cursor has been inserted in Container |
| |
| and P_Positions_Shifted |
| (Positions (Container)'Old, |
| Positions (Container), |
| Cut => Find (Elements (Container), New_Item)); |
| |
| procedure Include |
| (Container : in out Set; |
| New_Item : Element_Type) |
| with |
| Global => null, |
| Pre => |
| Length (Container) < Container.Capacity |
| or Contains (Container, New_Item), |
| Post => Contains (Container, New_Item), |
| Contract_Cases => |
| |
| -- If New_Item is already in Container |
| |
| (Contains (Container, New_Item) => |
| |
| -- Elements are preserved |
| |
| Model (Container)'Old = Model (Container) |
| |
| -- Cursors are preserved |
| |
| and Positions (Container) = Positions (Container)'Old |
| |
| -- The element equivalent to New_Item in Container is replaced by |
| -- New_Item. |
| |
| and E.Get (Elements (Container), |
| Find (Elements (Container), New_Item)) = New_Item |
| |
| and E.Equal_Except |
| (Elements (Container)'Old, |
| Elements (Container), |
| Find (Elements (Container), New_Item)), |
| |
| -- Otherwise, New_Item is inserted in Container |
| |
| others => |
| Length (Container) = Length (Container)'Old + 1 |
| |
| -- Other elements are preserved |
| |
| and Model (Container)'Old <= Model (Container) |
| and M.Included_Except |
| (Model (Container), |
| Model (Container)'Old, |
| New_Item) |
| |
| -- New_Item is inserted in Container |
| |
| and E.Get (Elements (Container), |
| Find (Elements (Container), New_Item)) = New_Item |
| |
| -- The Elements of Container located before New_Item are preserved |
| |
| and E.Range_Equal |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => 1, |
| Lst => Find (Elements (Container), New_Item) - 1) |
| |
| -- Other Elements are shifted by 1 |
| |
| and E.Range_Shifted |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => Find (Elements (Container), New_Item), |
| Lst => Length (Container)'Old, |
| Offset => 1) |
| |
| -- A new cursor has been inserted in Container |
| |
| and P_Positions_Shifted |
| (Positions (Container)'Old, |
| Positions (Container), |
| Cut => Find (Elements (Container), New_Item))); |
| |
| procedure Replace |
| (Container : in out Set; |
| New_Item : Element_Type) |
| with |
| Global => null, |
| Pre => Contains (Container, New_Item), |
| Post => |
| |
| -- Elements are preserved |
| |
| Model (Container)'Old = Model (Container) |
| |
| -- Cursors are preserved |
| |
| and Positions (Container) = Positions (Container)'Old |
| |
| -- The element equivalent to New_Item in Container is replaced by |
| -- New_Item. |
| |
| and E.Get (Elements (Container), |
| Find (Elements (Container), New_Item)) = New_Item |
| and E.Equal_Except |
| (Elements (Container)'Old, |
| Elements (Container), |
| Find (Elements (Container), New_Item)); |
| |
| procedure Exclude |
| (Container : in out Set; |
| Item : Element_Type) |
| with |
| Global => null, |
| Post => not Contains (Container, Item), |
| Contract_Cases => |
| |
| -- If Item is not in Container, nothing is changed |
| |
| (not Contains (Container, Item) => |
| Model (Container) = Model (Container)'Old |
| and Elements (Container) = Elements (Container)'Old |
| and Positions (Container) = Positions (Container)'Old, |
| |
| -- Otherwise, Item is removed from Container |
| |
| others => |
| Length (Container) = Length (Container)'Old - 1 |
| |
| -- Other elements are preserved |
| |
| and Model (Container) <= Model (Container)'Old |
| and M.Included_Except |
| (Model (Container)'Old, |
| Model (Container), |
| Item) |
| |
| -- The elements of Container located before Item are preserved |
| |
| and E.Range_Equal |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => 1, |
| Lst => Find (Elements (Container), Item)'Old - 1) |
| |
| -- The elements located after Item are shifted by 1 |
| |
| and E.Range_Shifted |
| (Left => Elements (Container), |
| Right => Elements (Container)'Old, |
| Fst => Find (Elements (Container), Item)'Old, |
| Lst => Length (Container), |
| Offset => 1) |
| |
| -- A cursor has been removed from Container |
| |
| and P_Positions_Shifted |
| (Positions (Container), |
| Positions (Container)'Old, |
| Cut => Find (Elements (Container), Item)'Old)); |
| |
| procedure Delete |
| (Container : in out Set; |
| Item : Element_Type) |
| with |
| Global => null, |
| Pre => Contains (Container, Item), |
| Post => |
| Length (Container) = Length (Container)'Old - 1 |
| |
| -- Item is no longer in Container |
| |
| and not Contains (Container, Item) |
| |
| -- Other elements are preserved |
| |
| and Model (Container) <= Model (Container)'Old |
| and M.Included_Except |
| (Model (Container)'Old, |
| Model (Container), |
| Item) |
| |
| -- The elements of Container located before Item are preserved |
| |
| and E.Range_Equal |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => 1, |
| Lst => Find (Elements (Container), Item)'Old - 1) |
| |
| -- The elements located after Item are shifted by 1 |
| |
| and E.Range_Shifted |
| (Left => Elements (Container), |
| Right => Elements (Container)'Old, |
| Fst => Find (Elements (Container), Item)'Old, |
| Lst => Length (Container), |
| Offset => 1) |
| |
| -- A cursor has been removed from Container |
| |
| and P_Positions_Shifted |
| (Positions (Container), |
| Positions (Container)'Old, |
| Cut => Find (Elements (Container), Item)'Old); |
| |
| procedure Delete |
| (Container : in out Set; |
| Position : in out Cursor) |
| with |
| Global => null, |
| Depends => (Container =>+ Position, Position => null), |
| Pre => Has_Element (Container, Position), |
| Post => |
| Position = No_Element |
| and Length (Container) = Length (Container)'Old - 1 |
| |
| -- The element at position Position is no longer in Container |
| |
| and not Contains (Container, Element (Container, Position)'Old) |
| and not P.Has_Key (Positions (Container), Position'Old) |
| |
| -- Other elements are preserved |
| |
| and Model (Container) <= Model (Container)'Old |
| and M.Included_Except |
| (Model (Container)'Old, |
| Model (Container), |
| Element (Container, Position)'Old) |
| |
| -- The elements of Container located before Position are preserved. |
| |
| and E.Range_Equal |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => 1, |
| Lst => P.Get (Positions (Container)'Old, Position'Old) - 1) |
| |
| -- The elements located after Position are shifted by 1 |
| |
| and E.Range_Shifted |
| (Left => Elements (Container), |
| Right => Elements (Container)'Old, |
| Fst => P.Get (Positions (Container)'Old, Position'Old), |
| Lst => Length (Container), |
| Offset => 1) |
| |
| -- Position has been removed from Container |
| |
| and P_Positions_Shifted |
| (Positions (Container), |
| Positions (Container)'Old, |
| Cut => P.Get (Positions (Container)'Old, Position'Old)); |
| |
| procedure Delete_First (Container : in out Set) with |
| Global => null, |
| Contract_Cases => |
| (Length (Container) = 0 => Length (Container) = 0, |
| others => |
| Length (Container) = Length (Container)'Old - 1 |
| |
| -- The first element has been removed from Container |
| |
| and not Contains (Container, First_Element (Container)'Old) |
| |
| -- Other elements are preserved |
| |
| and Model (Container) <= Model (Container)'Old |
| and M.Included_Except |
| (Model (Container)'Old, |
| Model (Container), |
| First_Element (Container)'Old) |
| |
| -- Other elements are shifted by 1 |
| |
| and E.Range_Shifted |
| (Left => Elements (Container), |
| Right => Elements (Container)'Old, |
| Fst => 1, |
| Lst => Length (Container), |
| Offset => 1) |
| |
| -- First has been removed from Container |
| |
| and P_Positions_Shifted |
| (Positions (Container), |
| Positions (Container)'Old, |
| Cut => 1)); |
| |
| procedure Delete_Last (Container : in out Set) with |
| Global => null, |
| Contract_Cases => |
| (Length (Container) = 0 => Length (Container) = 0, |
| others => |
| Length (Container) = Length (Container)'Old - 1 |
| |
| -- The last element has been removed from Container |
| |
| and not Contains (Container, Last_Element (Container)'Old) |
| |
| -- Other elements are preserved |
| |
| and Model (Container) <= Model (Container)'Old |
| and M.Included_Except |
| (Model (Container)'Old, |
| Model (Container), |
| Last_Element (Container)'Old) |
| |
| -- Others elements of Container are preserved |
| |
| and E.Range_Equal |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => 1, |
| Lst => Length (Container)) |
| |
| -- Last cursor has been removed from Container |
| |
| and Positions (Container) <= Positions (Container)'Old); |
| |
| procedure Union (Target : in out Set; Source : Set) with |
| Global => null, |
| Pre => |
| Length (Source) - Length (Target and Source) <= |
| Target.Capacity - Length (Target), |
| Post => |
| Length (Target) = Length (Target)'Old |
| - M.Num_Overlaps (Model (Target)'Old, Model (Source)) |
| + Length (Source) |
| |
| -- Elements already in Target are still in Target |
| |
| and Model (Target)'Old <= Model (Target) |
| |
| -- Elements of Source are included in Target |
| |
| and Model (Source) <= Model (Target) |
| |
| -- Elements of Target come from either Source or Target |
| |
| and |
| M.Included_In_Union |
| (Model (Target), Model (Source), Model (Target)'Old) |
| |
| -- Actual value of elements come from either Left or Right |
| |
| and |
| E_Elements_Included |
| (Elements (Target), |
| Model (Target)'Old, |
| Elements (Target)'Old, |
| Elements (Source)) |
| and |
| E_Elements_Included |
| (Elements (Target)'Old, Model (Target)'Old, Elements (Target)) |
| and |
| E_Elements_Included |
| (Elements (Source), |
| Model (Target)'Old, |
| Elements (Source), |
| Elements (Target)) |
| |
| -- Mapping from cursors of Target to elements is preserved |
| |
| and Mapping_Preserved |
| (E_Left => Elements (Target)'Old, |
| E_Right => Elements (Target), |
| P_Left => Positions (Target)'Old, |
| P_Right => Positions (Target)); |
| |
| function Union (Left, Right : Set) return Set with |
| Global => null, |
| Pre => Length (Left) <= Count_Type'Last - Length (Right), |
| Post => |
| Length (Union'Result) = Length (Left) |
| - M.Num_Overlaps (Model (Left), Model (Right)) |
| + Length (Right) |
| |
| -- Elements of Left and Right are in the result of Union |
| |
| and Model (Left) <= Model (Union'Result) |
| and Model (Right) <= Model (Union'Result) |
| |
| -- Elements of the result of union come from either Left or Right |
| |
| and |
| M.Included_In_Union |
| (Model (Union'Result), Model (Left), Model (Right)) |
| |
| -- Actual value of elements come from either Left or Right |
| |
| and |
| E_Elements_Included |
| (Elements (Union'Result), |
| Model (Left), |
| Elements (Left), |
| Elements (Right)) |
| and |
| E_Elements_Included |
| (Elements (Left), Model (Left), Elements (Union'Result)) |
| and |
| E_Elements_Included |
| (Elements (Right), |
| Model (Left), |
| Elements (Right), |
| Elements (Union'Result)); |
| |
| function "or" (Left, Right : Set) return Set renames Union; |
| |
| procedure Intersection (Target : in out Set; Source : Set) with |
| Global => null, |
| Post => |
| Length (Target) = |
| M.Num_Overlaps (Model (Target)'Old, Model (Source)) |
| |
| -- Elements of Target were already in Target |
| |
| and Model (Target) <= Model (Target)'Old |
| |
| -- Elements of Target are in Source |
| |
| and Model (Target) <= Model (Source) |
| |
| -- Elements both in Source and Target are in the intersection |
| |
| and |
| M.Includes_Intersection |
| (Model (Target), Model (Source), Model (Target)'Old) |
| |
| -- Actual value of elements of Target is preserved |
| |
| and E_Elements_Included (Elements (Target), Elements (Target)'Old) |
| and |
| E_Elements_Included |
| (Elements (Target)'Old, Model (Source), Elements (Target)) |
| |
| -- Mapping from cursors of Target to elements is preserved |
| |
| and Mapping_Preserved |
| (E_Left => Elements (Target), |
| E_Right => Elements (Target)'Old, |
| P_Left => Positions (Target), |
| P_Right => Positions (Target)'Old); |
| |
| function Intersection (Left, Right : Set) return Set with |
| Global => null, |
| Post => |
| Length (Intersection'Result) = |
| M.Num_Overlaps (Model (Left), Model (Right)) |
| |
| -- Elements in the result of Intersection are in Left and Right |
| |
| and Model (Intersection'Result) <= Model (Left) |
| and Model (Intersection'Result) <= Model (Right) |
| |
| -- Elements both in Left and Right are in the result of Intersection |
| |
| and |
| M.Includes_Intersection |
| (Model (Intersection'Result), Model (Left), Model (Right)) |
| |
| -- Actual value of elements come from Left |
| |
| and |
| E_Elements_Included |
| (Elements (Intersection'Result), Elements (Left)) |
| and |
| E_Elements_Included |
| (Elements (Left), Model (Right), Elements (Intersection'Result)); |
| |
| function "and" (Left, Right : Set) return Set renames Intersection; |
| |
| procedure Difference (Target : in out Set; Source : Set) with |
| Global => null, |
| Post => |
| Length (Target) = Length (Target)'Old - |
| M.Num_Overlaps (Model (Target)'Old, Model (Source)) |
| |
| -- Elements of Target were already in Target |
| |
| and Model (Target) <= Model (Target)'Old |
| |
| -- Elements of Target are not in Source |
| |
| and M.No_Overlap (Model (Target), Model (Source)) |
| |
| -- Elements in Target but not in Source are in the difference |
| |
| and |
| M.Included_In_Union |
| (Model (Target)'Old, Model (Target), Model (Source)) |
| |
| -- Actual value of elements of Target is preserved |
| |
| and E_Elements_Included (Elements (Target), Elements (Target)'Old) |
| and |
| E_Elements_Included |
| (Elements (Target)'Old, Model (Target), Elements (Target)) |
| |
| -- Mapping from cursors of Target to elements is preserved |
| |
| and Mapping_Preserved |
| (E_Left => Elements (Target), |
| E_Right => Elements (Target)'Old, |
| P_Left => Positions (Target), |
| P_Right => Positions (Target)'Old); |
| |
| function Difference (Left, Right : Set) return Set with |
| Global => null, |
| Post => |
| Length (Difference'Result) = Length (Left) - |
| M.Num_Overlaps (Model (Left), Model (Right)) |
| |
| -- Elements of the result of Difference are in Left |
| |
| and Model (Difference'Result) <= Model (Left) |
| |
| -- Elements of the result of Difference are in Right |
| |
| and M.No_Overlap (Model (Difference'Result), Model (Right)) |
| |
| -- Elements in Left but not in Right are in the difference |
| |
| and |
| M.Included_In_Union |
| (Model (Left), Model (Difference'Result), Model (Right)) |
| |
| -- Actual value of elements come from Left |
| |
| and |
| E_Elements_Included (Elements (Difference'Result), Elements (Left)) |
| and |
| E_Elements_Included |
| (Elements (Left), |
| Model (Difference'Result), |
| Elements (Difference'Result)); |
| |
| function "-" (Left, Right : Set) return Set renames Difference; |
| |
| procedure Symmetric_Difference (Target : in out Set; Source : Set) with |
| Global => null, |
| Pre => |
| Length (Source) - Length (Target and Source) <= |
| Target.Capacity - Length (Target) + Length (Target and Source), |
| Post => |
| Length (Target) = Length (Target)'Old - |
| 2 * M.Num_Overlaps (Model (Target)'Old, Model (Source)) + |
| Length (Source) |
| |
| -- Elements of the difference were not both in Source and in Target |
| |
| and M.Not_In_Both (Model (Target), Model (Target)'Old, Model (Source)) |
| |
| -- Elements in Target but not in Source are in the difference |
| |
| and |
| M.Included_In_Union |
| (Model (Target)'Old, Model (Target), Model (Source)) |
| |
| -- Elements in Source but not in Target are in the difference |
| |
| and |
| M.Included_In_Union |
| (Model (Source), Model (Target), Model (Target)'Old) |
| |
| -- Actual value of elements come from either Left or Right |
| |
| and |
| E_Elements_Included |
| (Elements (Target), |
| Model (Target)'Old, |
| Elements (Target)'Old, |
| Elements (Source)) |
| and |
| E_Elements_Included |
| (Elements (Target)'Old, Model (Target), Elements (Target)) |
| and |
| E_Elements_Included |
| (Elements (Source), Model (Target), Elements (Target)); |
| |
| function Symmetric_Difference (Left, Right : Set) return Set with |
| Global => null, |
| Pre => Length (Left) <= Count_Type'Last - Length (Right), |
| Post => |
| Length (Symmetric_Difference'Result) = Length (Left) - |
| 2 * M.Num_Overlaps (Model (Left), Model (Right)) + |
| Length (Right) |
| |
| -- Elements of the difference were not both in Left and Right |
| |
| and |
| M.Not_In_Both |
| (Model (Symmetric_Difference'Result), Model (Left), Model (Right)) |
| |
| -- Elements in Left but not in Right are in the difference |
| |
| and |
| M.Included_In_Union |
| (Model (Left), Model (Symmetric_Difference'Result), Model (Right)) |
| |
| -- Elements in Right but not in Left are in the difference |
| |
| and |
| M.Included_In_Union |
| (Model (Right), Model (Symmetric_Difference'Result), Model (Left)) |
| |
| -- Actual value of elements come from either Left or Right |
| |
| and |
| E_Elements_Included |
| (Elements (Symmetric_Difference'Result), |
| Model (Left), |
| Elements (Left), |
| Elements (Right)) |
| and |
| E_Elements_Included |
| (Elements (Left), |
| Model (Symmetric_Difference'Result), |
| Elements (Symmetric_Difference'Result)) |
| and |
| E_Elements_Included |
| (Elements (Right), |
| Model (Symmetric_Difference'Result), |
| Elements (Symmetric_Difference'Result)); |
| |
| function "xor" (Left, Right : Set) return Set |
| renames Symmetric_Difference; |
| |
| function Overlap (Left, Right : Set) return Boolean with |
| Global => null, |
| Post => |
| Overlap'Result = not (M.No_Overlap (Model (Left), Model (Right))); |
| |
| function Is_Subset (Subset : Set; Of_Set : Set) return Boolean with |
| Global => null, |
| Post => Is_Subset'Result = (Model (Subset) <= Model (Of_Set)); |
| |
| function First (Container : Set) return Cursor with |
| Global => null, |
| Contract_Cases => |
| (Length (Container) = 0 => |
| First'Result = No_Element, |
| |
| others => |
| Has_Element (Container, First'Result) |
| and P.Get (Positions (Container), First'Result) = 1); |
| |
| function First_Element (Container : Set) return Element_Type with |
| Global => null, |
| Pre => not Is_Empty (Container), |
| Post => |
| First_Element'Result = E.Get (Elements (Container), 1) |
| and E_Smaller_Than_Range |
| (Elements (Container), |
| 2, |
| Length (Container), |
| First_Element'Result); |
| |
| function Last (Container : Set) return Cursor with |
| Global => null, |
| Contract_Cases => |
| (Length (Container) = 0 => |
| Last'Result = No_Element, |
| |
| others => |
| Has_Element (Container, Last'Result) |
| and P.Get (Positions (Container), Last'Result) = |
| Length (Container)); |
| |
| function Last_Element (Container : Set) return Element_Type with |
| Global => null, |
| Pre => not Is_Empty (Container), |
| Post => |
| Last_Element'Result = E.Get (Elements (Container), Length (Container)) |
| and E_Bigger_Than_Range |
| (Elements (Container), |
| 1, |
| Length (Container) - 1, |
| Last_Element'Result); |
| |
| function Next (Container : Set; Position : Cursor) return Cursor with |
| Global => null, |
| Pre => |
| Has_Element (Container, Position) or else Position = No_Element, |
| Contract_Cases => |
| (Position = No_Element |
| or else P.Get (Positions (Container), Position) = Length (Container) |
| => |
| Next'Result = No_Element, |
| |
| others => |
| Has_Element (Container, Next'Result) |
| and then P.Get (Positions (Container), Next'Result) = |
| P.Get (Positions (Container), Position) + 1); |
| |
| procedure Next (Container : Set; Position : in out Cursor) with |
| Global => null, |
| Pre => |
| Has_Element (Container, Position) or else Position = No_Element, |
| Contract_Cases => |
| (Position = No_Element |
| or else P.Get (Positions (Container), Position) = Length (Container) |
| => |
| Position = No_Element, |
| |
| others => |
| Has_Element (Container, Position) |
| and then P.Get (Positions (Container), Position) = |
| P.Get (Positions (Container), Position'Old) + 1); |
| |
| function Previous (Container : Set; Position : Cursor) return Cursor with |
| Global => null, |
| Pre => |
| Has_Element (Container, Position) or else Position = No_Element, |
| Contract_Cases => |
| (Position = No_Element |
| or else P.Get (Positions (Container), Position) = 1 |
| => |
| Previous'Result = No_Element, |
| |
| others => |
| Has_Element (Container, Previous'Result) |
| and then P.Get (Positions (Container), Previous'Result) = |
| P.Get (Positions (Container), Position) - 1); |
| |
| procedure Previous (Container : Set; Position : in out Cursor) with |
| Global => null, |
| Pre => |
| Has_Element (Container, Position) or else Position = No_Element, |
| Contract_Cases => |
| (Position = No_Element |
| or else P.Get (Positions (Container), Position) = 1 |
| => |
| Position = No_Element, |
| |
| others => |
| Has_Element (Container, Position) |
| and then P.Get (Positions (Container), Position) = |
| P.Get (Positions (Container), Position'Old) - 1); |
| |
| function Find (Container : Set; Item : Element_Type) return Cursor with |
| Global => null, |
| Contract_Cases => |
| |
| -- If Item is not contained in Container, Find returns No_Element |
| |
| (not Contains (Model (Container), Item) => |
| not P.Has_Key (Positions (Container), Find'Result) |
| and Find'Result = No_Element, |
| |
| -- Otherwise, Find returns a valid cursor in Container |
| |
| others => |
| P.Has_Key (Positions (Container), Find'Result) |
| and P.Get (Positions (Container), Find'Result) = |
| Find (Elements (Container), Item) |
| |
| -- The element designated by the result of Find is Item |
| |
| and Equivalent_Elements |
| (Element (Container, Find'Result), Item)); |
| |
| function Floor (Container : Set; Item : Element_Type) return Cursor with |
| Global => null, |
| Contract_Cases => |
| (Length (Container) = 0 or else Item < First_Element (Container) => |
| Floor'Result = No_Element, |
| others => |
| Has_Element (Container, Floor'Result) |
| and |
| not (Item < E.Get (Elements (Container), |
| P.Get (Positions (Container), Floor'Result))) |
| and E_Is_Find |
| (Elements (Container), |
| Item, |
| P.Get (Positions (Container), Floor'Result))); |
| |
| function Ceiling (Container : Set; Item : Element_Type) return Cursor with |
| Global => null, |
| Contract_Cases => |
| (Length (Container) = 0 or else Last_Element (Container) < Item => |
| Ceiling'Result = No_Element, |
| others => |
| Has_Element (Container, Ceiling'Result) |
| and |
| not (E.Get (Elements (Container), |
| P.Get (Positions (Container), Ceiling'Result)) < |
| Item) |
| and E_Is_Find |
| (Elements (Container), |
| Item, |
| P.Get (Positions (Container), Ceiling'Result))); |
| |
| function Contains (Container : Set; Item : Element_Type) return Boolean with |
| Global => null, |
| Post => Contains'Result = Contains (Model (Container), Item); |
| pragma Annotate (GNATprove, Inline_For_Proof, Contains); |
| |
| function Has_Element (Container : Set; Position : Cursor) return Boolean |
| with |
| Global => null, |
| Post => |
| Has_Element'Result = P.Has_Key (Positions (Container), Position); |
| pragma Annotate (GNATprove, Inline_For_Proof, Has_Element); |
| |
| generic |
| type Key_Type (<>) is private; |
| |
| with function Key (Element : Element_Type) return Key_Type; |
| |
| with function "<" (Left, Right : Key_Type) return Boolean is <>; |
| |
| package Generic_Keys with SPARK_Mode is |
| |
| function Equivalent_Keys (Left, Right : Key_Type) return Boolean with |
| Global => null, |
| Post => |
| Equivalent_Keys'Result = (not (Left < Right) and not (Right < Left)); |
| pragma Annotate (GNATprove, Inline_For_Proof, Equivalent_Keys); |
| |
| package Formal_Model with Ghost is |
| function E_Bigger_Than_Range |
| (Container : E.Sequence; |
| Fst : Positive_Count_Type; |
| Lst : Count_Type; |
| Key : Key_Type) return Boolean |
| with |
| Global => null, |
| Pre => Lst <= E.Length (Container), |
| Post => |
| E_Bigger_Than_Range'Result = |
| (for all I in Fst .. Lst => |
| Generic_Keys.Key (E.Get (Container, I)) < Key); |
| pragma Annotate (GNATprove, Inline_For_Proof, E_Bigger_Than_Range); |
| |
| function E_Smaller_Than_Range |
| (Container : E.Sequence; |
| Fst : Positive_Count_Type; |
| Lst : Count_Type; |
| Key : Key_Type) return Boolean |
| with |
| Global => null, |
| Pre => Lst <= E.Length (Container), |
| Post => |
| E_Smaller_Than_Range'Result = |
| (for all I in Fst .. Lst => |
| Key < Generic_Keys.Key (E.Get (Container, I))); |
| pragma Annotate (GNATprove, Inline_For_Proof, E_Smaller_Than_Range); |
| |
| function E_Is_Find |
| (Container : E.Sequence; |
| Key : Key_Type; |
| Position : Count_Type) return Boolean |
| with |
| Global => null, |
| Pre => Position - 1 <= E.Length (Container), |
| Post => |
| E_Is_Find'Result = |
| |
| ((if Position > 0 then |
| E_Bigger_Than_Range (Container, 1, Position - 1, Key)) |
| |
| and (if Position < E.Length (Container) then |
| E_Smaller_Than_Range |
| (Container, |
| Position + 1, |
| E.Length (Container), |
| Key))); |
| pragma Annotate (GNATprove, Inline_For_Proof, E_Is_Find); |
| |
| function Find |
| (Container : E.Sequence; |
| Key : Key_Type) return Count_Type |
| -- Search for Key in Container |
| |
| with |
| Global => null, |
| Post => |
| (if Find'Result > 0 then |
| Find'Result <= E.Length (Container) |
| and Equivalent_Keys |
| (Key, Generic_Keys.Key (E.Get (Container, Find'Result))) |
| and E_Is_Find (Container, Key, Find'Result)); |
| |
| function M_Included_Except |
| (Left : M.Set; |
| Right : M.Set; |
| Key : Key_Type) return Boolean |
| with |
| Global => null, |
| Post => |
| M_Included_Except'Result = |
| (for all E of Left => |
| Contains (Right, E) |
| or Equivalent_Keys (Generic_Keys.Key (E), Key)); |
| end Formal_Model; |
| use Formal_Model; |
| |
| function Key (Container : Set; Position : Cursor) return Key_Type with |
| Global => null, |
| Post => Key'Result = Key (Element (Container, Position)); |
| pragma Annotate (GNATprove, Inline_For_Proof, Key); |
| |
| function Element (Container : Set; Key : Key_Type) return Element_Type |
| with |
| Global => null, |
| Pre => Contains (Container, Key), |
| Post => |
| Element'Result = Element (Container, Find (Container, Key)); |
| pragma Annotate (GNATprove, Inline_For_Proof, Element); |
| |
| procedure Replace |
| (Container : in out Set; |
| Key : Key_Type; |
| New_Item : Element_Type) |
| with |
| Global => null, |
| Pre => Contains (Container, Key), |
| Post => |
| Length (Container) = Length (Container)'Old |
| |
| -- Key now maps to New_Item |
| |
| and Element (Container, Key) = New_Item |
| |
| -- New_Item is contained in Container |
| |
| and Contains (Model (Container), New_Item) |
| |
| -- Other elements are preserved |
| |
| and M_Included_Except |
| (Model (Container)'Old, |
| Model (Container), |
| Key) |
| and M.Included_Except |
| (Model (Container), |
| Model (Container)'Old, |
| New_Item) |
| |
| -- Mapping from cursors to elements is preserved |
| |
| and Mapping_Preserved_Except |
| (E_Left => Elements (Container)'Old, |
| E_Right => Elements (Container), |
| P_Left => Positions (Container)'Old, |
| P_Right => Positions (Container), |
| Position => Find (Container, Key)) |
| and Positions (Container) = Positions (Container)'Old; |
| |
| procedure Exclude (Container : in out Set; Key : Key_Type) with |
| Global => null, |
| Post => not Contains (Container, Key), |
| Contract_Cases => |
| |
| -- If Key is not in Container, nothing is changed |
| |
| (not Contains (Container, Key) => |
| Model (Container) = Model (Container)'Old |
| and Elements (Container) = Elements (Container)'Old |
| and Positions (Container) = Positions (Container)'Old, |
| |
| -- Otherwise, Key is removed from Container |
| |
| others => |
| Length (Container) = Length (Container)'Old - 1 |
| |
| -- Other elements are preserved |
| |
| and Model (Container) <= Model (Container)'Old |
| and M_Included_Except |
| (Model (Container)'Old, |
| Model (Container), |
| Key) |
| |
| -- The elements of Container located before Key are preserved |
| |
| and E.Range_Equal |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => 1, |
| Lst => Find (Elements (Container), Key)'Old - 1) |
| |
| -- The elements located after Key are shifted by 1 |
| |
| and E.Range_Shifted |
| (Left => Elements (Container), |
| Right => Elements (Container)'Old, |
| Fst => Find (Elements (Container), Key)'Old, |
| Lst => Length (Container), |
| Offset => 1) |
| |
| -- A cursor has been removed from Container |
| |
| and P_Positions_Shifted |
| (Positions (Container), |
| Positions (Container)'Old, |
| Cut => Find (Elements (Container), Key)'Old)); |
| |
| procedure Delete (Container : in out Set; Key : Key_Type) with |
| Global => null, |
| Pre => Contains (Container, Key), |
| Post => |
| Length (Container) = Length (Container)'Old - 1 |
| |
| -- Key is no longer in Container |
| |
| and not Contains (Container, Key) |
| |
| -- Other elements are preserved |
| |
| and Model (Container) <= Model (Container)'Old |
| and M_Included_Except |
| (Model (Container)'Old, |
| Model (Container), |
| Key) |
| |
| -- The elements of Container located before Key are preserved |
| |
| and E.Range_Equal |
| (Left => Elements (Container)'Old, |
| Right => Elements (Container), |
| Fst => 1, |
| Lst => Find (Elements (Container), Key)'Old - 1) |
| |
| -- The elements located after Key are shifted by 1 |
| |
| and E.Range_Shifted |
| (Left => Elements (Container), |
| Right => Elements (Container)'Old, |
| Fst => Find (Elements (Container), Key)'Old, |
| Lst => Length (Container), |
| Offset => 1) |
| |
| -- A cursor has been removed from Container |
| |
| and P_Positions_Shifted |
| (Positions (Container), |
| Positions (Container)'Old, |
| Cut => Find (Elements (Container), Key)'Old); |
| |
| function Find (Container : Set; Key : Key_Type) return Cursor with |
| Global => null, |
| Contract_Cases => |
| |
| -- If Key is not contained in Container, Find returns No_Element |
| |
| ((for all E of Model (Container) => |
| not Equivalent_Keys (Key, Generic_Keys.Key (E))) => |
| not P.Has_Key (Positions (Container), Find'Result) |
| and Find'Result = No_Element, |
| |
| -- Otherwise, Find returns a valid cursor in Container |
| |
| others => |
| P.Has_Key (Positions (Container), Find'Result) |
| and P.Get (Positions (Container), Find'Result) = |
| Find (Elements (Container), Key) |
| |
| -- The element designated by the result of Find is Key |
| |
| and Equivalent_Keys |
| (Generic_Keys.Key (Element (Container, Find'Result)), Key)); |
| |
| function Floor (Container : Set; Key : Key_Type) return Cursor with |
| Global => null, |
| Contract_Cases => |
| (Length (Container) = 0 |
| or else Key < Generic_Keys.Key (First_Element (Container)) => |
| Floor'Result = No_Element, |
| others => |
| Has_Element (Container, Floor'Result) |
| and |
| not (Key < |
| Generic_Keys.Key |
| (E.Get (Elements (Container), |
| P.Get (Positions (Container), Floor'Result)))) |
| and E_Is_Find |
| (Elements (Container), |
| Key, |
| P.Get (Positions (Container), Floor'Result))); |
| |
| function Ceiling (Container : Set; Key : Key_Type) return Cursor with |
| Global => null, |
| Contract_Cases => |
| (Length (Container) = 0 |
| or else Generic_Keys.Key (Last_Element (Container)) < Key => |
| Ceiling'Result = No_Element, |
| others => |
| Has_Element (Container, Ceiling'Result) |
| and |
| not (Generic_Keys.Key |
| (E.Get (Elements (Container), |
| P.Get (Positions (Container), Ceiling'Result))) |
| < Key) |
| and E_Is_Find |
| (Elements (Container), |
| Key, |
| P.Get (Positions (Container), Ceiling'Result))); |
| |
| function Contains (Container : Set; Key : Key_Type) return Boolean with |
| Global => null, |
| Post => |
| Contains'Result = |
| (for some E of Model (Container) => |
| Equivalent_Keys (Key, Generic_Keys.Key (E))); |
| |
| end Generic_Keys; |
| |
| private |
| pragma SPARK_Mode (Off); |
| |
| pragma Inline (Next); |
| pragma Inline (Previous); |
| |
| type Node_Type is record |
| Has_Element : Boolean := False; |
| Parent : Count_Type := 0; |
| Left : Count_Type := 0; |
| Right : Count_Type := 0; |
| Color : Red_Black_Trees.Color_Type; |
| Element : aliased Element_Type; |
| end record; |
| |
| package Tree_Types is |
| new Red_Black_Trees.Generic_Bounded_Tree_Types (Node_Type); |
| |
| type Set (Capacity : Count_Type) is record |
| Content : Tree_Types.Tree_Type (Capacity); |
| end record; |
| |
| use Red_Black_Trees; |
| |
| Empty_Set : constant Set := (Capacity => 0, others => <>); |
| |
| end Ada.Containers.Formal_Ordered_Sets; |
