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FreeRTOS-Kernel/verification/verifast/proof/verifast_lock_predicates.h

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#ifndef VERIFAST_LOCK_PREDICATES_H
#define VERIFAST_LOCK_PREDICATES_H
#include "verifast_task_running_states.h"
/* We follow a minimalistic approach during the definition of the
* lock predicates. So far, the only encapsulate the resources and
* invariants required to verify `vTaskSwitchContext`.
* We are going to extend and refine them when we proceed to verify
* other parts of FRTOS.
*/
#include "verifast_lists_extended.h"
/* ----------------------------------------------------------------------
* Core local data and access restrictions.
* Some data in FreeRTOS such as the pointer to TCB of the task running
* on core `C` may only be accessed from core `C`. Such core-local data
* protected by deactivating interrupts.
*/
/*@
predicate interruptState_p(uint32_t coreID, uint32_t state);
fixpoint bool interruptsDisabled_f(uint32_t);
predicate coreLocalInterruptInv_p() =
[0.5]pointer(&pxCurrentTCBs[coreID_f], ?currentTCB) &*&
//pubTCB_p(currentTCB, 0) &*&
integer_(&xYieldPendings[coreID_f], sizeof(BaseType_t), true, _) &*&
coreLocalSeg_TCB_p(currentTCB, ?gCriticalNesting);
@*/
/* ----------------------------------------------------------------------
* Predicates relevant for all locks
*/
/*@
predicate locked_p(list< pair<real, int> > lockHistory);
@*/
/* ----------------------------------------------------------------------
* Task lock and associated global variables from `tasks.c`
*/
/*@
fixpoint int taskLockID_f();
// Represents an acquired task lock.
predicate taskLock_p();
// Represents the invariant associated with the the task lock, i.e.,
// access permissions to the resources protected by the lock.
predicate taskLockInv_p();
@*/
/* ----------------------------------------------------------------------
* ISR lock and associated global variables from `tasks.c`
*/
/*@
fixpoint int isrLockID_f();
// Represents an unacquired ISR lock.
predicate isrLock_p();
// Represents the invariant associated with the the ISR lock, i.e.,
// access permissions to the resources protected by the lock.
predicate isrLockInv_p();
@*/
/* ----------------------------------------------------------------------
* Resources protected by both locks.
* Note that the task lock may never be acquired after the ISR lock.
*/
/*@
fixpoint int taskISRLockID_f();
predicate taskISRLockInv_p() =
// Access to global variables
[0.5]pointer(&pxCurrentTCBs[coreID_f], ?gCurrentTCB) &*&
integer_((void*) &uxSchedulerSuspended, sizeof(UBaseType_t), false, _) &*&
integer_(&xSchedulerRunning, sizeof(BaseType_t), true, _)
&*&
// top ready priority must be in range
integer_((void*) &uxTopReadyPriority, sizeof(UBaseType_t), false, ?gTopReadyPriority) &*&
0 <= gTopReadyPriority &*& gTopReadyPriority < configMAX_PRIORITIES
&*&
// tasks / TCBs
exists_in_taskISRLockInv_p(?gTasks, ?gStates)
&*&
// (RP-All) Read permissions for every task
// and recording of task states in state list
// (∀t ∈ gTasks.
// [1/2]sharedSeg_TCB_p(t, _))
// ∧
// ∀i. ∀t. gTasks[i] == t -> gStates[i] == t->xTaskRunState
foreach(gTasks, readOnly_sharedSeg_TCB_p(gTasks, gStates))
&*&
// (RP-Current) Read permission for task currently scheduled on this core
// (RP-All) + (RP-Current) => Write permission for scheduled task
[1/2]sharedSeg_TCB_p(gCurrentTCB, ?gCurrentTCB_state) &*&
// gCurrentTCB_state != taskTASK_NOT_RUNNING &*&
(gCurrentTCB_state == coreID_f() || gCurrentTCB_state == taskTASK_YIELDING) &*&
nth(index_of(gCurrentTCB, gTasks), gStates) == gCurrentTCB_state
&*&
// (RP-Unsched) Read permissions for unscheduled tasks
// (RP-All) + (RP-Unsched) => Write permissions for unscheduled tasks
// ∀t ∈ tasks. t->xTaskState == taskTASK_NOT_RUNNING
// -> [1/2]shared_TCB_p(t, taskTASK_NOT_RUNNING)
foreach(gTasks, readOnly_sharedSeg_TCB_IF_not_running_p(gTasks, gStates))
&*&
readyLists_p(?gCellLists, ?gOwnerLists)
&*&
// gTasks contains all relevant tasks
mem(gCurrentTCB, gTasks) == true
&*&
// ∀l ∈ gOwnerLists. l ⊆ gTasks
forall(gOwnerLists, (superset)(gTasks)) == true;
// Auxiliary predicate. Equal to the lock invariant above but exposes
// some details.
predicate _taskISRLockInv_p(UBaseType_t gTopReadyPriority) =
// Access to global variables
[0.5]pointer(&pxCurrentTCBs[coreID_f], ?gCurrentTCB) &*&
integer_((void*) &uxSchedulerSuspended, sizeof(UBaseType_t), false, _) &*&
integer_(&xSchedulerRunning, sizeof(BaseType_t), true, _)
&*&
// top ready priority must be in range
integer_((void*) &uxTopReadyPriority, sizeof(UBaseType_t), false, gTopReadyPriority) &*&
0 <= gTopReadyPriority &*& gTopReadyPriority < configMAX_PRIORITIES
&*&
// tasks / TCBs
exists_in_taskISRLockInv_p(?gTasks, ?gStates)
&*&
// (RP-All) Read permissions for every task
// and recording of task states in state list
// (∀t ∈ gTasks.
// [1/2]sharedSeg_TCB_p(t, _))
// ∧
// ∀i. ∀t. gTasks[i] == t -> gStates[i] == t->xTaskRunState
foreach(gTasks, readOnly_sharedSeg_TCB_p(gTasks, gStates))
&*&
// (RP-Current) Read permission for task currently scheduled on this core
// (RP-All) + (RP-Current) => Write permission for scheduled task
[1/2]sharedSeg_TCB_p(gCurrentTCB, ?gCurrentTCB_state) &*&
// gCurrentTCB_state != taskTASK_NOT_RUNNING &*&
(gCurrentTCB_state == coreID_f() || gCurrentTCB_state == taskTASK_YIELDING) &*&
nth(index_of(gCurrentTCB, gTasks), gStates) == gCurrentTCB_state
&*&
// (RP-Unsched) Read permissions for unscheduled tasks
// (RP-All) + (RP-Unsched) => Write permissions for unscheduled tasks
// ∀t ∈ tasks. t->xTaskState == taskTASK_NOT_RUNNING
// -> [1/2]shared_TCB_p(t, taskTASK_NOT_RUNNING)
foreach(gTasks, readOnly_sharedSeg_TCB_IF_not_running_p(gTasks, gStates))
&*&
readyLists_p(?gCellLists, ?gOwnerLists)
&*&
// gTasks contains all relevant tasks
mem(gCurrentTCB, gTasks) == true
&*&
// ∀l ∈ gOwnerLists. l ⊆ gTasks
forall(gOwnerLists, (superset)(gTasks)) == true;
lemma void produce_taskISRLockInv();
requires locked_p(?heldLocks) &*&
heldLocks == cons(?i, cons(?t, nil)) &*&
i == pair(?f_isr, isrLockID_f()) &*&
t == pair(?f_task, taskLockID_f());
ensures locked_p( cons( pair(_, taskISRLockID_f()), heldLocks) ) &*&
taskISRLockInv_p();
lemma void consume_taskISRLockInv();
requires locked_p( cons( pair(_, taskISRLockID_f()), ?otherLocks) ) &*&
taskISRLockInv_p();
ensures locked_p(otherLocks);
// Auxiliary predicate to assing names to existentially quantified variables.
// Having multiple `exists` chunks on the heap makes matching against their
// arguments ambiguous in most cases.
predicate exists_in_taskISRLockInv_p(list<void*> gTasks,
list<TaskRunning_t> gStates) =
exists(gTasks) &*&
exists(gStates) &*&
length(gTasks) == length(gStates) &*&
distinct(gTasks) == true;
// Auxiliary function that allows us to partially apply the list argument.
//
// Notes:
// - Partial application of fixpoint functions in VeriFast is not documented.
// The syntax for partially application is `(<fixpoint_fct>)(<first_arg>)`
// - VeriFast only supports partially applying the first argument, e.g.,
// `(mem)(0)` is allowed but `(mem)(_)(nil)` is not.
fixpoint bool mem_list_elem<t>(list<t> xs, t x) {
return mem(x, xs);
}
// Auxiliary predicate to allow foreach-quantification about fraction
// and reflection of `t->xTaskRunState` in state list.
predicate_ctor readOnly_sharedSeg_TCB_p
(list<void*> tasks, list<TaskRunning_t> states)
(TCB_t* t;) =
mem(t, tasks) == true &*&
[1/2]sharedSeg_TCB_p(t, nth(index_of(t, tasks), states));
predicate_ctor readOnly_sharedSeg_TCB_IF_not_running_p
(list<void*> tasks, list<TaskRunning_t> states)
(TCB_t* t;) =
mem(t, tasks) == true &*&
nth(index_of(t, tasks), states) == taskTASK_NOT_RUNNING
? [1/2]sharedSeg_TCB_p(t, taskTASK_NOT_RUNNING)
: true;
@*/
/*@
lemma void nonauto_nth_update<t>(int i, int j, t y, list<t> xs);
requires 0 <= i && i < length(xs) && 0 <= j && j < length(xs);
ensures nth(i, update(j, y, xs)) == (i == j ? y : nth(i, xs));
@*/
// -----------------------------------------------------------------------
// TODO: Move lemmas below to separate header file.
/*@
lemma void update_readOnly_sharedSeg_TCB(TCB_t* t,
list<void*> tasks,
list<TaskRunning_t> states,
int updatedIndex,
TaskRunning_t s)
requires readOnly_sharedSeg_TCB_p(tasks, states)(t) &*&
updatedIndex != index_of(t, tasks) &*&
mem(t, tasks) == true &*&
length(tasks) == length(states);
ensures readOnly_sharedSeg_TCB_p(tasks, update(updatedIndex, s, states))(t);
{
list<TaskRunning_t> states2 = update(updatedIndex, s, states);
int t_index = index_of(t, tasks);
if( updatedIndex < 0 || updatedIndex >= length(states) ) {
update_out_of_bounds(updatedIndex, s, states);
} else {
open readOnly_sharedSeg_TCB_p(tasks, states)(t);
open [1/2]sharedSeg_TCB_p(t, nth(t_index, states));
mem_index_of(t, tasks);
nth_update(t_index, updatedIndex, s, states);
assert( nth(t_index, states) == nth(t_index, states2) );
close [1/2]sharedSeg_TCB_p(t, nth(t_index, states2));
close readOnly_sharedSeg_TCB_p(tasks, states2)(t);
}
}
lemma void update_foreach_readOnly_sharedSeg_TCB(TCB_t* updatedTask,
list<void*> tasks,
list<void*> subTasks,
list<TaskRunning_t> states,
list<TaskRunning_t> states2,
TaskRunning_t s)
requires
mem(updatedTask, tasks) == true &*&
length(tasks) == length(states) &*&
foreach(subTasks, readOnly_sharedSeg_TCB_p(tasks, states)) &*&
states2 == update(index_of(updatedTask, tasks), s, states) &*&
distinct(tasks) == true &*&
mem(updatedTask, subTasks) == false &*&
subset(subTasks, tasks) == true;
ensures
foreach(subTasks, readOnly_sharedSeg_TCB_p(tasks, states2));
{
switch(subTasks) {
case nil:
open foreach(nil, readOnly_sharedSeg_TCB_p(tasks, states));
close foreach(nil, readOnly_sharedSeg_TCB_p(tasks, states2));
case cons(h, rest):
int index = index_of(updatedTask, tasks);
// distinct_mem_remove(t, tasks);
// neq_mem_remove(h, t, tasks);
// index_of_different(h, t, tasks);
open foreach(subTasks, readOnly_sharedSeg_TCB_p(tasks, states));
assert( updatedTask != h );
index_of_different(updatedTask, h, tasks);
assert( index != index_of(h, tasks) );
update_readOnly_sharedSeg_TCB(h, tasks, states, index, s);
assert( mem(updatedTask, rest) == false );
update_foreach_readOnly_sharedSeg_TCB(updatedTask, tasks, rest,
states, states2, s);
close foreach(subTasks, readOnly_sharedSeg_TCB_p(tasks, states2));
}
}
lemma void close_updated_foreach_readOnly_sharedSeg_TCB(TCB_t* updatedTask,
list<void*> tasks,
list<TaskRunning_t> states,
list<TaskRunning_t> states2,
TaskRunning_t s)
requires
mem(updatedTask, tasks) == true &*&
length(states) == length(tasks) &*&
distinct(tasks) == true &*&
foreach(remove(updatedTask, tasks), readOnly_sharedSeg_TCB_p(tasks, states)) &*&
states2 == update(index_of(updatedTask, tasks), s, states) &*&
[1/2]sharedSeg_TCB_p(updatedTask, s);
ensures
foreach(tasks, readOnly_sharedSeg_TCB_p(tasks, states2));
{
distinct_mem_remove(updatedTask, tasks);
remove_result_subset(updatedTask, tasks);
close readOnly_sharedSeg_TCB_p(tasks, states2)(updatedTask);
update_foreach_readOnly_sharedSeg_TCB(updatedTask, tasks,
remove(updatedTask, tasks),
states, states2, s);
foreach_unremove(updatedTask, tasks);
}
lemma void stopUpdate_foreach_readOnly_sharedSeg_TCB_IF_not_running
(TCB_t* stoppedTask, list<void*> tasks, list<void*> subTasks,
list<TaskRunning_t> states, list<TaskRunning_t> states2)
requires
distinct(tasks) == true &*&
distinct(subTasks) == true &*&
length(tasks) == length(states) &*&
foreach(subTasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states)) &*&
states2 == update(index_of(stoppedTask, tasks), taskTASK_NOT_RUNNING, states) &*&
nth(index_of(stoppedTask, tasks), states) != taskTASK_NOT_RUNNING &*&
subset(subTasks, tasks) == true &*&
mem(stoppedTask, tasks) == true &*&
mem(stoppedTask, subTasks)
? [1/2]sharedSeg_TCB_p(stoppedTask, taskTASK_NOT_RUNNING)
: true;
ensures
foreach(subTasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states2));
{
switch(subTasks) {
case nil:
open foreach(nil, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states));
close foreach(nil, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states2));
case cons(h, t):
if( h == stoppedTask ) {
assert( remove(stoppedTask, subTasks) == t );
distinct_mem_remove(stoppedTask, subTasks);
assert( mem(stoppedTask, t) == false );
mem_index_of(stoppedTask, tasks);
} else {
mem_index_of(stoppedTask, tasks);
nth_update(index_of(h, tasks), index_of(stoppedTask, tasks), taskTASK_NOT_RUNNING, states);
index_of_different(h, stoppedTask, tasks);
assert( index_of(h, tasks) != index_of(stoppedTask, tasks) );
assert( nth(index_of(h, tasks), states) == nth(index_of(h, tasks), states2) );
}
nth_update(index_of(stoppedTask, tasks), index_of(stoppedTask, tasks), taskTASK_NOT_RUNNING, states);
open foreach(subTasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states));
open readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states)(h);
assert( nth(index_of(stoppedTask, tasks), states2) == taskTASK_NOT_RUNNING );
close readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states2)(h);
stopUpdate_foreach_readOnly_sharedSeg_TCB_IF_not_running
(stoppedTask, tasks, t, states, states2);
close foreach(subTasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states2));
}
}
lemma void updateUnaffectedStates_in_foreach_readOnly_sharedSeg_TCB_IF_not_running
(TCB_t* updatedTask, list<void*> tasks, list<void*> subTasks,
list<TaskRunning_t> states, list<TaskRunning_t> updatedStates,
TaskRunning_t s)
requires
distinct(tasks) == true &*&
distinct(subTasks) == true &*&
length(tasks) == length(states) &*&
mem(updatedTask, tasks) == true &*&
mem(updatedTask, subTasks) == false &*&
subset(subTasks, tasks) == true &*&
foreach(subTasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states)) &*&
updatedStates == update(index_of(updatedTask, tasks), s, states);
ensures
foreach(subTasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, updatedStates));
{
switch(subTasks) {
case nil:
open foreach(nil, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states));
close foreach(nil, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, updatedStates));
case cons(h, t):
open foreach(subTasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states));
// Prove that update preserves state of `h`.
index_of_different(h, updatedTask, tasks);
nth_update(index_of(h, tasks), index_of(updatedTask, tasks), s, states);
assert( nth(index_of(h, tasks), states) == nth(index_of(h, tasks), updatedStates) );
open readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states)(h);
close readOnly_sharedSeg_TCB_IF_not_running_p(tasks, updatedStates)(h);
updateUnaffectedStates_in_foreach_readOnly_sharedSeg_TCB_IF_not_running
(updatedTask, tasks, t, states, updatedStates, s);
close foreach(subTasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, updatedStates));
}
}
lemma void startUpdate_foreach_readOnly_sharedSeg_TCB_IF_not_running
(TCB_t* startedTask, list<void*> tasks,
list<TaskRunning_t> states, list<TaskRunning_t> updatedStates,
int coreID)
requires
distinct(tasks) == true &*&
length(tasks) == length(states) &*&
mem(startedTask, tasks) == true &*&
foreach(remove(startedTask, tasks), readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states)) &*&
updatedStates == update(index_of(startedTask, tasks), coreID, states) &*&
0 <= coreID &*& coreID < configNUM_CORES;
ensures
foreach(tasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, updatedStates));
{
distinct_remove(startedTask, tasks);
distinct_mem_remove(startedTask, tasks);
remove_result_subset(startedTask, tasks);
updateUnaffectedStates_in_foreach_readOnly_sharedSeg_TCB_IF_not_running
(startedTask, tasks, remove(startedTask, tasks), states, updatedStates,
coreID);
assert( foreach(remove(startedTask, tasks), readOnly_sharedSeg_TCB_IF_not_running_p(tasks, updatedStates)) );
close readOnly_sharedSeg_TCB_IF_not_running_p(tasks, updatedStates)(startedTask);
foreach_unremove(startedTask, tasks);
}
lemma void scheduleRunning_in_foreach_readOnly_sharedSeg_TCB_IF_not_running
(TCB_t* runningTask, list<void*> tasks,
list<TaskRunning_t> states, list<TaskRunning_t> updatedStates,
int coreID)
requires
distinct(tasks) == true &*&
length(tasks) == length(states) &*&
mem(runningTask, tasks) == true &*&
(nth(index_of(runningTask, tasks), states) == coreID
|| nth(index_of(runningTask, tasks), states) == taskTASK_YIELDING)
&*&
foreach(tasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states)) &*&
updatedStates == update(index_of(runningTask, tasks), coreID, states) &*&
0 <= coreID &*& coreID < configNUM_CORES;
ensures
foreach(tasks, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, updatedStates)) &*&
nth(index_of(runningTask, tasks), updatedStates) == coreID;
{
switch(tasks) {
case nil:
open foreach(nil, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states));
close foreach(nil, readOnly_sharedSeg_TCB_IF_not_running_p(tasks, updatedStates));
case cons(h, t):
foreach_remove(runningTask, tasks);
distinct_remove(runningTask, tasks);
distinct_mem_remove(runningTask, tasks);
remove_result_subset(runningTask, tasks);
updateUnaffectedStates_in_foreach_readOnly_sharedSeg_TCB_IF_not_running
(runningTask, tasks, remove(runningTask, tasks),
states, updatedStates, coreID);
open readOnly_sharedSeg_TCB_IF_not_running_p(tasks, states)(runningTask);
close readOnly_sharedSeg_TCB_IF_not_running_p(tasks, updatedStates)(runningTask);
foreach_unremove(runningTask, tasks);
}
}
@*/
/*@
lemma list<TaskRunning_t> def_state1(list<void*> tasks,
list<TaskRunning_t> states,
TCB_t* currentTask,
TCB_t* readyTask)
requires
distinct(tasks) == true &*&
length(tasks) == length(states) &*&
currentTask != readyTask &*&
mem(currentTask, tasks) == true &*&
mem(readyTask, tasks) == true &*&
nth(index_of(readyTask, tasks), states) == taskTASK_NOT_RUNNING;
ensures
result == update(index_of(currentTask, tasks), taskTASK_NOT_RUNNING, states) &*&
nth(index_of(readyTask, tasks), result) == taskTASK_NOT_RUNNING &*&
nth(index_of(currentTask, tasks), result) == taskTASK_NOT_RUNNING;
{
list<TaskRunning_t> states1 =
update(index_of(currentTask, tasks), taskTASK_NOT_RUNNING, states);
mem_index_of(currentTask, tasks);
mem_index_of(readyTask, tasks);
nth_update(index_of(readyTask, tasks), index_of(currentTask, tasks), taskTASK_NOT_RUNNING, states);
return states1;
}
lemma list<TaskRunning_t> def_state2(list<void*> tasks,
list<TaskRunning_t> states,
TCB_t* currentTask,
TCB_t* readyTask,
int coreID)
requires
distinct(tasks) == true &*&
length(tasks) == length(states) &*&
currentTask != readyTask &*&
mem(currentTask, tasks) == true &*&
mem(readyTask, tasks) == true &*&
nth(index_of(readyTask, tasks), states) == taskTASK_NOT_RUNNING &*&
nth(index_of(currentTask, tasks), states) != taskTASK_NOT_RUNNING &*&
0 <= coreID &*& coreID < configNUM_CORES;
ensures
result ==
update(index_of(readyTask, tasks), coreID,
update(index_of(currentTask, tasks), taskTASK_NOT_RUNNING, states))
&*&
nth(index_of(readyTask, tasks), result) == coreID &*&
nth(index_of(currentTask, tasks), result) == taskTASK_NOT_RUNNING;
{
list<TaskRunning_t> states1 = def_state1(tasks, states, currentTask, readyTask);
list<TaskRunning_t> states2 =
update(index_of(readyTask, tasks), coreID, states1);
return states2;
}
@*/
#endif /* VERIFAST_LOCK_PREDICATES_H */
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