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SatSolverMain.v
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(**********************************************************************************
* The PEDANTIC (Proof Engine for Deductive Automation using Non-deterministic
* Traversal of Instruction Code) verification framework
*
* Developed by Kenneth Roe
* For more information, check out www.cs.jhu.edu/~roe
*
* TreeTraversal.v
* This file contains the proof of correctness of a tree traversal algorithm using
* the PEDANTIC verification framework.
*
**********************************************************************************)
Require Export SfLib.
Require Export SfLibExtras.
Require Export ImpHeap.
Require Export AbsState.
Require Export AbsExecute.
Require Export AbsStateInstance.
Require Export Simplify.
Require Export Eqdep.
Require Export StateImplication.
Require Export Classical.
Require Export Unfold.
Require Export Fold.
Require Export merge.
Require Export ProgramTactics.
Require Export SatSolverDefs.
Require Export SatSolverAux1.
Opaque basicEval.
Opaque invariant.
Opaque haveVarInvariant.
Definition x := 1.
Theorem varNeqBacktrack : varx <> backtrack.
Proof.
admit.
Admitted.
Theorem validRefTheorem4 : forall s n b, id -> nat -> realizeState (AbsUpdateVar invariant assignments_to_do_head !!(todo))
nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b todo) :: @NatValue unit todo_var_offset :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem5 : forall s n b, id -> nat -> realizeState
(AbsUpdateLoc
(AbsUpdateVar invariant assignments_to_do_head !!(todo))
!!(todo) ++++ #todo_var_offset !!(varx)) nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b todo) :: @NatValue unit todo_val_offset :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem7: forall s n b, id -> nat -> realizeState
(AbsUpdateVar
([!!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
([!!(assignments_to_do_tail)] ** invariant)
varx !!(assignments_to_do_tail) ++++ #todo_var_offset)
valuex !!(assignments_to_do_tail) ++++ #todo_val_offset)
prop !!(assignments_to_do_tail) ++++ #todo_unit_offset)
ssss !!(assignments_to_do_tail) ++++ #prev_offset)
assignments_to_do_tail !!(ssss)) nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b ssss) :: @NatValue unit next_offset :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem existsTheorem1 : exists s : state,
realizeState
(AbsMagicWand
(AbsUpdateLoc
(AbsUpdateVar
([!!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
([!!(assignments_to_do_tail)] ** invariant)
varx
!!(assignments_to_do_tail) ++++ #todo_var_offset)
valuex
!!(assignments_to_do_tail) ++++ #todo_val_offset)
prop !!(assignments_to_do_tail) ++++ #todo_unit_offset)
ssss !!(assignments_to_do_tail) ++++ #prev_offset)
assignments_to_do_tail !!(ssss)) !!(ssss) ++++ #next_offset
#0)
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #4 |-> v(5) **
v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) ** [!!(ssss) ==== v(0)]))))))))
nil s.
Proof.
admit.
Admitted.
Theorem existsTheorem2 : exists s : state,
realizeState
(AbsMagicWand
([~~ !!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
([!!(assignments_to_do_tail)] ** invariant)
varx !!(assignments_to_do_tail) ++++ #todo_var_offset)
valuex !!(assignments_to_do_tail) ++++ #todo_val_offset)
prop !!(assignments_to_do_tail) ++++ #todo_unit_offset)
ssss !!(assignments_to_do_tail) ++++ #prev_offset)
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #4 |-> v(5) **
v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) ** [!!(ssss) ==== v(0)]))))))))
nil s.
Proof.
admit.
Admitted.
Theorem mergeTheorem4 : mergeStates
(AbsMagicWand
(AbsUpdateLoc
(AbsUpdateVar
([!!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
([!!(assignments_to_do_tail)] ** invariant)
varx
!!(assignments_to_do_tail) ++++ #todo_var_offset)
valuex
!!(assignments_to_do_tail) ++++ #todo_val_offset)
prop !!(assignments_to_do_tail) ++++ #todo_unit_offset)
ssss !!(assignments_to_do_tail) ++++ #prev_offset)
assignments_to_do_tail !!(ssss)) !!(ssss) ++++ #next_offset
#0)
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #4 |-> v(5) **
v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) ** [!!(ssss) ==== v(0)]))))))))
(AbsUpdateVar
(AbsUpdateVar
(AbsMagicWand
([~~ !!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
([!!(assignments_to_do_tail)] ** invariant)
varx
!!(assignments_to_do_tail) ++++ #todo_var_offset)
valuex
!!(assignments_to_do_tail) ++++ #todo_val_offset)
prop !!(assignments_to_do_tail) ++++ #todo_unit_offset)
ssss !!(assignments_to_do_tail) ++++ #prev_offset)
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #4 |-> v(5) **
v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) **
[!!(ssss) ==== v(0)]))))))))
assignments_to_do_head #0) assignments_to_do_tail
#0) invariant.
Proof.
admit.
Admitted.
Theorem existsTheorem3 : exists s : state,
realizeState
(AbsMagicWand
(AbsUpdateWithLoc ([!!(assignments_to_do_head)] ** invariantNoTail)
todo !!(assignments_to_do_head) ++++ #next_offset)
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #4 |-> v(5) **
v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) **
[!!(assignments_to_do_head) ==== v(0)]))))))))
nil s.
Proof.
admit.
Admitted.
Theorem entailment3 : forall s : state,
realizeState
(AbsUpdateVar
(AbsMagicWand
(AbsUpdateWithLoc
([!!(assignments_to_do_head)] ** invariantNoTail)
todo !!(assignments_to_do_head) ++++ #next_offset)
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #4 |-> v(5) **
v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) **
[!!(assignments_to_do_head) ==== v(0)]))))))))
assignments_to_do_head !!(todo)) nil s ->
realizeState invariantNoTail nil s.
Proof.
admit.
Admitted.
Theorem entailment4 : forall x0 : state,
realizeState
([~~ !!(ssss) ==== !!(valuex)] **
[!!(ssss)] **
AbsUpdateWithLoc invariant ssss !!(assignments) ++++ !!(varx)) nil x0 ->
realizeState invariantNoTail nil x0.
Proof.
admit.
Admitted.
Theorem validRefTheorem8 : forall s n b, id -> nat -> realizeState
(AbsUpdateWithLoc
(AbsUpdateWithLoc
([!!(stack) //\\ (!!(ssss) \\// !!(vvvv) ==== #2)] ** invariant)
kkkk !!(stack) ++++ #next_offset) vvvv
!!(stack) ++++ #stack_var_offset) nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b assignments) :: @NatValue unit (b vvvv) :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem existsTheorem4 : exists s : state,
realizeState
(AbsMagicWand
(AbsUpdateLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
([!!(stack) //\\ (!!(ssss) \\// !!(vvvv) ==== #2)] **
invariant) kkkk !!(stack) ++++ #next_offset)
vvvv !!(stack) ++++ #stack_var_offset)
!!(assignments) ++++ !!(vvvv) #0)
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) ** [!!(stack) ==== v(0)])))))))
nil s.
Proof.
admit.
Admitted.
Theorem entailment5: forall s : state,
realizeState
(AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateVar
(AbsMagicWand
(AbsUpdateLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc
([!!(stack) //\\ (!!(ssss) \\// !!(vvvv) ==== #2)] **
invariant) kkkk !!(stack) ++++ #next_offset)
vvvv !!(stack) ++++ #stack_var_offset)
!!(assignments) ++++ !!(vvvv) #0)
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) **
[!!(stack) ==== v(0)])))))))
stack !!(kkkk)) ssss !!(stack) ++++ #stack_prop_offset)
vvvv !!(stack) ++++ #stack_val_offset) nil s ->
realizeState invariant nil s.
Proof.
admit.
Admitted.
Theorem entailment6 : forall x0 : state,
realizeState
(AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateVar ([~~ !!(assignments_to_do_head)] ** invariantNoTail)
assignments_to_do_tail #0) ssss
!!(stack) ++++ #stack_prop_offset) vvvv
!!(stack) ++++ #stack_val_offset) nil x0 ->
realizeState invariant nil x0.
Proof.
admit.
Admitted.
Theorem validRefTheorem9 : forall s n b, id -> nat -> realizeState
([~~ !!(stack) ==== #0] **
[~~ (convertToAbsExp (ALand (!stack) (ALor (!ssss) (!vvvv === A2))))] **
invariant) nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b stack) :: @NatValue unit stack_val_offset :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem10 : forall s n b, id -> nat -> realizeState
(AbsUpdateWithLoc
(AbsUpdateLoc
([~~ !!(stack) ==== #0] **
[~~
(convertToAbsExp
(ALand (!stack) (ALor (!ssss) (!vvvv === A2))))] **
invariant) !!(stack) ++++ #stack_val_offset
#2) vvvv !!(stack) ++++ #stack_var_offset) nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b assignments) :: @NatValue unit (b vvvv) :: nil) -> heap_p s n <> None.
Proof.
intros s n b H H0.
admit.
Admitted.
Theorem mergeReturn1 : mergeReturnStates invariant
([!!(stack) ==== #0] **
[~~ (convertToAbsExp (ALand (!stack) (ALor (!ssss) (!vvvv === A2))))] **
invariant) invariant (#0 :: #1 :: nil) (#0 :: nil)
(#0 :: #1 :: nil).
Proof.
admit.
Admitted.
Theorem mergeReturn2 : mergeReturnStates invariant invariant invariant
(#0 :: #1 :: nil) (#0 :: #1 :: nil) (#0 :: #1 :: nil).
Proof.
admit.
Admitted.
Theorem mergeTheorem5 : mergeStates
([!!(ssss) ==== !!(valuex)] **
[!!(ssss)] **
AbsUpdateWithLoc invariant ssss !!(assignments) ++++ !!(varx))
(AbsUpdateVar
(AbsUpdateLoc
(AbsUpdateWithLoc
(AbsUpdateLoc
([~~ !!(stack) ==== #0] **
[~~
(convertToAbsExp
(ALand (!stack) (ALor (!ssss) (!vvvv === A2))))] **
invariant) !!(stack) ++++ #stack_val_offset
#2) vvvv !!(stack) ++++ #stack_var_offset)
!!(assignments) ++++ !!(vvvv) #2) backtrack
#1) invariant.
Proof.
admit.
Admitted.
Theorem validRefTheorem11 : forall s n b, id -> nat -> realizeState
([~~ !!(ssss)] **
AbsUpdateWithLoc invariant ssss !!(assignments) ++++ !!(varx)) nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b assignments) :: @NatValue unit (b varx) :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem12 : forall s n b, id -> nat -> realizeState
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) **
[!!(ssss) ==== v(0)] **
AbsExistsT
(AbsUpdateLoc
([~~ v(5)] **
AbsUpdateWithLoc
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(quantifyAbsVarState invariant
ssss))))))
ssss !!(assignments) ++++ !!(varx))
!!(assignments) ++++ !!(varx)
!!(valuex)))))))) nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b ssss) :: @NatValue unit next_offset :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem13 : forall s n b, id -> nat -> realizeState
(AbsUpdateVar
(AbsUpdateLoc
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) **
[!!(ssss) ==== v(0)] **
AbsExistsT
(AbsUpdateLoc
([~~ v(5)] **
AbsUpdateWithLoc
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(quantifyAbsVarState
invariant
ssss))))))
ssss !!(assignments) ++++ !!(varx))
!!(assignments) ++++ !!(varx)
!!(valuex))))))))
!!(ssss) ++++ #next_offset !!(stack))
stack !!(ssss)) nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b ssss) :: @NatValue unit stack_var_offset :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem14 : forall s n b, id -> nat -> realizeState
(AbsUpdateLoc
(AbsUpdateVar
(AbsUpdateLoc
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) **
[!!(ssss) ==== v(0)] **
AbsExistsT
(AbsUpdateLoc
([~~ v(5)] **
AbsUpdateWithLoc
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(quantifyAbsVarState
invariant
ssss))))))
ssss !!(assignments) ++++ !!(varx))
!!(assignments) ++++ !!(varx)
!!(valuex))))))))
!!(ssss) ++++ #next_offset !!(stack))
stack !!(ssss)) !!(ssss) ++++ #stack_var_offset
!!(varx)) nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b ssss) :: @NatValue unit stack_val_offset :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem15 : forall s n b, id -> nat -> realizeState
(AbsUpdateLoc
(AbsUpdateLoc
(AbsUpdateVar
(AbsUpdateLoc
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) **
[!!(ssss) ==== v(0)] **
AbsExistsT
(AbsUpdateLoc
([~~ v(5)] **
AbsUpdateWithLoc
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(quantifyAbsVarState
invariant
ssss))))))
ssss
!!(assignments) ++++ !!(varx))
!!(assignments) ++++ !!(varx)
!!(valuex))))))))
!!(ssss) ++++ #next_offset !!(stack))
stack !!(ssss)) !!(ssss) ++++ #stack_var_offset
!!(varx)) !!(ssss) ++++ #stack_val_offset
!!(valuex)) nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b ssss) :: @NatValue unit stack_prop_offset :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem mergeTheorem6 : mergeStates
(AbsUpdateVar
([!!(ssss) //\\ !!(vvvv) ==== #2] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateVar ([~~ #3 <<<< !!(jjjj)] ** invariantWLIT1)
jjjj !!(jjjj) ++++ #1) ssss
(!!(clause) ++++ #positive_lit_offset) ++++ !!(jjjj))
vvvv !!(assignments) ++++ !!(jjjj)) skip
#1)
([~~ (!!(ssss) //\\ !!(vvvv) ==== #2)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateVar ([~~ #3 <<<< !!(jjjj)] ** invariantWLIT1)
jjjj !!(jjjj) ++++ #1) ssss
(!!(clause) ++++ #positive_lit_offset) ++++ !!(jjjj))
vvvv !!(assignments) ++++ !!(jjjj)) invariantWLIT1.
Proof.
admit.
Admitted.
Theorem mergeTheorem7 : mergeStates
(AbsUpdateVar
([!!(ssss) //\\ !!(vvvv) ==== #1] **
AbsUpdateWithLoc invariantWLIT1 ssss
(!!(clause) ++++ #negative_lit_offset) ++++ !!(jjjj))
skip #1)
([~~ (!!(ssss) //\\ !!(vvvv) ==== #1)] **
AbsUpdateWithLoc invariantWLIT1 ssss
(!!(clause) ++++ #negative_lit_offset) ++++ !!(jjjj)) invariantWLIT1.
Proof.
admit.
Admitted.
Theorem mergeTheorem8 : mergeStates
(AbsUpdateVar
(AbsUpdateVar
([!!(vvvv) //\\ !!(ssss) ==== #0] **
AbsUpdateWithLoc invariantWLIT1 ssss
(!!(clause) ++++ #watch_var_offset) ++++ !!(jjjj))
non_watch !!(jjjj)) skip #1)
([~~ (!!(vvvv) //\\ !!(ssss) ==== #0)] **
AbsUpdateWithLoc invariantWLIT1 ssss
(!!(clause) ++++ #watch_var_offset) ++++ !!(jjjj)) invariantWLIT1.
Proof.
admit.
Admitted.
Theorem entailment7 : forall x0 : state,
realizeState
(AbsUpdateVar
(AbsUpdateVar
(AbsUpdateVar
([!!(valuex) ==== #2 //\\ !!(ssss) \\//
!!(valuex) ==== #1 //\\ !!(vvvv)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateWithLoc ([!!(clause)] ** invariantWL)
nc (!!(clause) ++++ #watch_next_offset) ++++ !!(varx))
ssss (!!(clause) ++++ #negative_lit_offset) ++++ !!(varx))
vvvv (!!(clause) ++++ #positive_lit_offset) ++++ !!(varx))
skip #0) skip #0) jjjj #0) nil x0 ->
realizeState invariantWLIT1 nil x0.
Proof.
admit.
Admitted.
Theorem validRefTheorem16 : forall s n b, id -> nat -> realizeState
(AbsUpdateWithLoc
([!!(skip)] **
[~~ !!(skip)] **
[~~ (convertToAbsExp (!jjjj <<= A3))] ** invariantWLIT1)
ssss !!(watches) ++++ !!(non_watch)) nil s -> NatValue n =
basicEval assignments_to_do_head
(basicEval assignments_to_do_head
(NatValue (b clause) :: NatValue watch_next_offset :: nil)
:: @NatValue unit (b non_watch) :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem17 : forall s n b, id -> nat -> realizeState
(AbsUpdateLoc
(AbsUpdateWithLoc
([!!(skip)] **
[~~ !!(skip)] **
[~~ (convertToAbsExp (!jjjj <<= A3))] ** invariantWLIT1)
ssss !!(watches) ++++ !!(non_watch))
(!!(clause) ++++ #watch_next_offset) ++++ !!(non_watch)
!!(ssss)) nil s -> NatValue n =
basicEval assignments_to_do_head
(basicEval assignments_to_do_head
(NatValue (b clause) :: NatValue watch_var_offset :: nil)
:: @NatValue unit (b non_watch) :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem18 : forall s n b, id -> nat -> realizeState
([!!(ssss)] **
AbsUpdateLoc
(AbsUpdateLoc
(AbsUpdateWithLoc
([!!(skip)] **
[~~ !!(skip)] **
[~~ (convertToAbsExp (!jjjj <<= A3))] ** invariantWLIT1)
ssss !!(watches) ++++ !!(non_watch))
(!!(clause) ++++ #watch_next_offset) ++++ !!(non_watch)
!!(ssss))
(!!(clause) ++++ #watch_var_offset) ++++ !!(non_watch)
#1) nil s -> NatValue n =
basicEval assignments_to_do_head
(basicEval assignments_to_do_head
(NatValue (b ssss) :: NatValue watch_prev_offset :: nil)
:: @NatValue unit (b non_watch) :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem mergeTheorem9 : mergeStates
(AbsUpdateLoc
([!!(ssss)] **
AbsUpdateLoc
(AbsUpdateLoc
(AbsUpdateWithLoc
([!!(skip)] **
[~~ !!(skip)] **
[~~ (convertToAbsExp (!jjjj <<= A3))] ** invariantWLIT1)
ssss !!(watches) ++++ !!(non_watch))
(!!(clause) ++++ #watch_next_offset) ++++ !!(non_watch)
!!(ssss))
(!!(clause) ++++ #watch_var_offset) ++++ !!(non_watch)
#1) (!!(ssss) ++++ #watch_prev_offset) ++++ !!(non_watch)
!!(clause))
([~~ !!(ssss)] **
AbsUpdateLoc
(AbsUpdateLoc
(AbsUpdateWithLoc
([!!(skip)] **
[~~ !!(skip)] **
[~~ (convertToAbsExp (!jjjj <<= A3))] ** invariantWLIT1)
ssss !!(watches) ++++ !!(non_watch))
(!!(clause) ++++ #watch_next_offset) ++++ !!(non_watch)
!!(ssss)) (!!(clause) ++++ #watch_var_offset) ++++ !!(non_watch)
#1) invariantWLIT1.
Proof.
admit.
Admitted.
Theorem validRefTheorem19 : forall s n b, id -> nat -> realizeState invariantWLIT1 nil s -> NatValue n =
basicEval assignments_to_do_head
(NatValue (b watches) :: @NatValue unit (b non_watch) :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem20 : forall s n b, id -> nat -> realizeState
(AbsUpdateLoc invariantWLIT1 !!(watches) ++++ !!(non_watch)
!!(clause)) nil s -> NatValue n =
basicEval assignments_to_do_head
(basicEval assignments_to_do_head
(NatValue (b clause) :: NatValue watch_var_offset :: nil)
:: @NatValue unit (b varx) :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem21 : forall s n b, id -> nat -> realizeState
([!!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateLoc
(AbsUpdateLoc invariantWLIT1 !!(watches) ++++ !!(non_watch)
!!(clause))
(!!(clause) ++++ #watch_var_offset) ++++ !!(varx)
#0) ssss (!!(clause) ++++ #watch_prev_offset) ++++ !!(varx))
vvvv (!!(clause) ++++ #watch_next_offset) ++++ !!(varx)) nil s -> NatValue n =
basicEval assignments_to_do_head
(basicEval assignments_to_do_head
(NatValue (b ssss) :: NatValue watch_next_offset :: nil)
:: @NatValue unit (b varx) :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem22 : forall s n b, id -> nat -> realizeState
([~~ !!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateLoc
(AbsUpdateLoc invariantWLIT1 !!(watches) ++++ !!(non_watch)
!!(clause))
(!!(clause) ++++ #watch_var_offset) ++++ !!(varx)
#0) ssss (!!(clause) ++++ #watch_prev_offset) ++++ !!(varx))
vvvv (!!(clause) ++++ #watch_next_offset) ++++ !!(varx)) nil s -> NatValue n =
basicEval assignments_to_do_head
(@NatValue unit (b watches) :: NatValue (b varx) :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem mergeTheorem10 : mergeStates
(AbsUpdateLoc
([!!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateLoc
(AbsUpdateLoc invariantWLIT1 !!(watches) ++++ !!(non_watch)
!!(clause))
(!!(clause) ++++ #watch_var_offset) ++++ !!(varx)
#0) ssss (!!(clause) ++++ #watch_prev_offset) ++++ !!(varx))
vvvv (!!(clause) ++++ #watch_next_offset) ++++ !!(varx))
(!!(ssss) ++++ #watch_next_offset) ++++ !!(varx)
!!(vvvv))
(AbsUpdateLoc
([~~ !!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc
(AbsUpdateLoc
(AbsUpdateLoc invariantWLIT1 !!(watches) ++++ !!(non_watch)
!!(clause))
(!!(clause) ++++ #watch_var_offset) ++++ !!(varx)
#0) ssss (!!(clause) ++++ #watch_prev_offset) ++++ !!(varx))
vvvv (!!(clause) ++++ #watch_next_offset) ++++ !!(varx))
!!(watches) ++++ !!(varx) !!(vvvv)) invariantWLIT1.
Proof.
admit.
Admitted.
Theorem validRefTheorem23 : forall s n b, id -> nat -> realizeState ([!!(vvvv)] ** invariantWLIT1) nil s -> NatValue n =
basicEval assignments_to_do_head
(basicEval assignments_to_do_head
(NatValue (b vvvv) :: NatValue watch_prev_offset :: nil)
:: @NatValue unit (b varx) :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem mergeTheorem11 : mergeStates
(AbsUpdateLoc ([!!(vvvv)] ** invariantWLIT1)
(!!(vvvv) ++++ #watch_prev_offset) ++++ !!(varx)
!!(ssss)) ([~~ !!(vvvv)] ** invariantWLIT1) invariantWLIT1.
Proof.
admit.
Admitted.
Theorem mergeTheorem12 : mergeStates
(AbsUpdateVar
([!!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateVar
([!!(ssss) //\\ !!(jjjj) ==== #0] **
AbsUpdateWithLoc
(AbsUpdateWithLoc ([~~ #3 <<<< !!(kkkk)] ** invariantWLIT2)
ssss (!!(clause) ++++ #watch_var_offset) ++++ !!(kkkk))
jjjj !!(assignments) ++++ !!(kkkk))
vvvv !!(kkkk)) ssss
(!!(clause) ++++ #positive_lit_offset) ++++ !!(vvvv))
val #2)
([~~ !!(ssss)] **
AbsUpdateWithLoc
(AbsUpdateVar
([!!(ssss) //\\ !!(jjjj) ==== #0] **
AbsUpdateWithLoc
(AbsUpdateWithLoc ([~~ #3 <<<< !!(kkkk)] ** invariantWLIT2)
ssss (!!(clause) ++++ #watch_var_offset) ++++ !!(kkkk))
jjjj !!(assignments) ++++ !!(kkkk)) vvvv
!!(kkkk)) ssss
(!!(clause) ++++ #positive_lit_offset) ++++ !!(vvvv)) invariantWLIT2.
Proof.
admit.
Admitted.
Theorem mergeTheorem13 : mergeStates
(AbsUpdateVar
([!!(ssss)] **
AbsUpdateWithLoc invariantWLIT2 ssss
(!!(clause) ++++ #negative_lit_offset) ++++ !!(vvvv))
val #1)
([~~ !!(ssss)] **
AbsUpdateWithLoc invariantWLIT2 ssss
(!!(clause) ++++ #negative_lit_offset) ++++ !!(vvvv)) invariantWLIT2.
Proof.
admit.
Admitted.
Theorem mergeTheorem14 : mergeStates invariantWLIT2
([~~ (!!(ssss) //\\ !!(jjjj) ==== #0)] **
AbsUpdateWithLoc
(AbsUpdateWithLoc ([~~ #3 <<<< !!(kkkk)] ** invariantWLIT2)
ssss (!!(clause) ++++ #watch_var_offset) ++++ !!(kkkk))
jjjj !!(assignments) ++++ !!(kkkk)) invariantWLIT2.
Proof.
admit.
Admitted.
Theorem entailment8 : forall s : state,
realizeState (AbsUpdateVar invariantWLIT2 kkkk !!(kkkk) ++++ #1) nil s ->
realizeState invariantWLIT2 nil s.
Proof.
admit.
Admitted.
Theorem entailment9 : forall x0 : state,
realizeState
(AbsUpdateVar
([~~ !!(skip)] **
[~~ !!(skip)] **
[~~ (convertToAbsExp (!jjjj <<= A3))] ** invariantWLIT1)
kkkk #0) nil x0 -> realizeState invariantWLIT2 nil x0.
Proof.
admit.
Admitted.
Theorem validRefTheorem24 : forall s n b, id -> nat -> realizeState
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #4 |-> v(5) **
v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) **
[!!(todo) ==== v(0)] **
AbsExistsT
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(quantifyAbsVarState
([
~~
(convertToAbsExp
(!(kkkk) <<= A3))] **
invariantWLIT2)
todo)))))))))))))) nil s -> NatValue n =
basicEval assignments_to_do_head
(@NatValue unit (b todo) :: NatValue next_offset :: nil) -> heap_p s n <> None.
Proof.
admit.
Admitted.
Theorem validRefTheorem25 : forall s n b, id -> nat -> realizeState
(AbsUpdateLoc
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(AbsExistsT
(v(0) ++++ #4 |-> v(5) **
v(0) ++++ #3 |-> v(4) **
v(0) ++++ #2 |-> v(3) **
v(0) ++++ #1 |-> v(2) **
v(0) ++++ #0 |-> v(1) **
[!!(todo) ==== v(0)] **
AbsExistsT
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState
(pushAbsVarState