\settings {
"#Proof-Settings-Config-File
#Sun Dec 07 16:22:56 EST 2014
[DLOptions]counterExampleGenerator=Mathematica
[DecisionProcedure]Exec=
[DLOptions]termFactoryClass=de.uka.ilkd.key.dl.model.impl.TermFactoryImpl
[Z3Options]z3ElimExPrefix=true
[ReduceOptions]qepcadFallback=false
[MathematicaOptions]eliminateFractions=false
[TacletTranslation]filename=
[StrategyProperty]USER_TACLETS_OPTIONS_KEY3=USER_TACLETS_OFF
[DLOptions]BuiltInArithmeticIneqs=OFF
[ReduceOptions]rlcadpbfvs=DEFAULT
[ReduceOptions]rlcadaproj=DEFAULT
[StrategyProperty]USER_TACLETS_OPTIONS_KEY2=USER_TACLETS_OFF
[StrategyProperty]USER_TACLETS_OPTIONS_KEY1=USER_TACLETS_OFF
[General]ProofAssistant=false
[ReduceOptions]rlcadprojonly=DEFAULT
[DLOptions]reduceOnFreshBranch=false
[StrategyProperty]LOOP_OPTIONS_KEY=LOOP_INVARIANT
[ReduceOptions]rlqeqsc=DEFAULT
[ReduceOptions]rlcadbaseonly=DEFAULT
[DLOptions]simplifyAfterReduce=false
[DLOptions]readdQuantifiers=true
[DecisionProcedure]ActiveRule=_noname_
[ReduceOptions]rlcadrawformula=DEFAULT
[DLOptions]applyLocalSimplify=false
[SimultaneousUpdateSimplifier]DeleteEffectLessLocations=true
[ReduceOptions]quantifierEliminationMethod=RLQE
[DLOptions]DiffSat=AUTO
[DLOptions]simplifier=Mathematica
[StrategyProperty]QUERY_OPTIONS_KEY=QUERY_NONE
[StrategyProperty]QUANTIFIERS_OPTIONS_KEY=QUANTIFIERS_NON_SPLITTING_WITH_PROGS
[DLOptions]simplifyAfterODESolve=false
[ReduceOptions]rlall=true
[MetiTarskiOptions]metitBacktracking=true
[DLOptions]diffSatTimeout=4000
[DLOptions]linearTimeoutIncreaseFactor=2
[ReduceOptions]groebnerBasis=false
[DecisionProcedure]Timeout=600
[ReduceOptions]rlqesqsc=DEFAULT
[General]SoundNotification=false
[StrategyProperty]STOPMODE_OPTIONS_KEY=STOPMODE_DEFAULT
[Choice]DefaultChoices=transactions-transactions\\:transactionsOn , throughout-throughout\\:toutOn , programRules-programRules\\:Java , intRules-intRules\\:arithmeticSemanticsIgnoringOF , initialisation-initialisation\\:disableStaticInitialisation , transactionAbort-transactionAbort\\:abortOn , nullPointerPolicy-nullPointerPolicy\\:nullCheck , javacard-javacard\\:jcOff
[ReduceOptions]rlsimpl=ON
[DLOptions]universalClosureOnQE=true
[MathematicaOptions]useParallelize=false
[ReduceOptions]rlcadextonly=DEFAULT
[Strategy]MaximumNumberOfAutomaticApplications=6000
[ReduceOptions]rlanuexsgnopt=DEFAULT
[ReduceOptions]rlcadpartial=DEFAULT
[General]UseJML=true
[DLOptions]useSOS=false
[MathematicaOptions]useEliminateList=true
[DecisionProcedure]savefile=false
[DLOptions]applyGammaRules=ONLY_TO_MODALITIES
[DecisionProcedure]showSMTResDialog=false
[TacletTranslation]assignment=11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
[DLOptions]QEOnlyToTrue=false
[General]StupidMode=true
[DLOptions]constantTimeoutIncreaseFactor=0
[DLOptions]solveODE=true
[StrategyProperty]METHOD_OPTIONS_KEY=METHOD_EXPAND
[DLOptions]loopSatTimeout=2000000
[DLOptions]odeSolver=Mathematica
[OrbitalOptions]precision=34
[HOLLightOptions]qeMethod=ProofProducing
[ReduceOptions]rlqedfs=DEFAULT
[HOLLightOptions]useSnapshots=false
[QepcadOptions]qepcadMemoryLimit=2000000
[ReduceOptions]rlqeheu=DEFAULT
[ReduceOptions]rlanuexpsremseq=DEFAULT
[DecisionProcedure]multprovers=Z3\\=true\\:Yices\\=true\\:Simplify\\=true\\:CVC3\\=true
[ReduceOptions]rlcadfulldimonly=DEFAULT
[DLOptions]csdpForceInternal=false
[DLOptions]invariantRule=FRESH
[DLOptions]addRigidFormulas=true
[General]DnDDirectionSensitive=true
[DLOptions]simplifyBeforeReduce=false
[Z3Options]z3Prenex=true
[DLOptions]applyToModality=false
[DLOptions]initialTimeout=2000
[ReduceOptions]rlcadaprojalways=DEFAULT
[DLOptions]CexFinder=ITER_DEEP
[DLOptions]useIterativeReduceRule=false
[Choice]Choices=throughout-throughout\\:toutOn-throughout\\:toutOff , transactions-transactions\\:transactionsOn-transactions\\:transactionsOff , intRules-intRules\\:javaSemantics-intRules\\:arithmeticSemanticsIgnoringOF-intRules\\:arithmeticSemanticsCheckingOF , programRules-programRules\\:Java-programRules\\:ODL-programRules\\:dL , initialisation-initialisation\\:enableStaticInitialisation-initialisation\\:disableStaticInitialisation , transactionAbort-transactionAbort\\:abortOn-transactionAbort\\:abortOff , nullPointerPolicy-nullPointerPolicy\\:noNullCheck-nullPointerPolicy\\:nullCheck , javacard-javacard\\:jcOff-javacard\\:jcOn
[CohenhormanderOptions]eliminatorMode=DNF
[DLOptions]percentOfPowersetForIterativeReduce=70
[hints]used=
[ReduceOptions]rlcadhongproj=DEFAULT
[ReduceOptions]rlanuexgcdnormalize=DEFAULT
[MathematicaOptions]memoryConstraint=-1
[ReduceOptions]rlcadtrimtree=DEFAULT
[DLOptions]groebnerBasisCalculator=Mathematica
[ReduceOptions]rlcadfac=DEFAULT
[Strategy]Timeout=60000
[DLOptions]applyGlobalReduce=true
[OrbitalOptions]representation=big
[SimultaneousUpdateSimplifier]EagerSimplification=true
[TacletTranslation]maxGeneric=2
[DLOptions]ibcOnlyToFO=true
[DLOptions]TracerStat=OFF
[TacletTranslation]saveToFile=false
[DLOptions]BuiltInArithmetic=APPLY_EQUALITIES
[DLOptions]counterexampleTest=ON
[StrategyProperty]GOALCHOOSER_OPTIONS_KEY=GOALCHOOSER_DEFAULT
[DecisionProcedure]WaitForAllProvers=false
[DLOptions]useODEIndFinMethods=false
[StrategyProperty]NON_LIN_ARITH_OPTIONS_KEY=NON_LIN_ARITH_NONE
[ReduceOptions]rlcadisoallroots=DEFAULT
[ReduceOptions]rlnzden=ON
[MathematicaOptions]quantifierEliminationMethod=REDUCE
[StrategyProperty]SPLITTING_OPTIONS_KEY=SPLITTING_DELAYED
[ReduceOptions]eliminateFractions=false
[DecisionProcedure]WeakenSMTTranslation=false
[ReduceOptions]rlcadte=DEFAULT
[DLOptions]ignoreAnnotations=false
[StrategyProperty]VBT_PHASE=VBT_SYM_EX
[DLOptions]sosChecker=Built-in SOS
[General]UseOCL=false
[OrbitalOptions]sparsePolynomials=true
[ReduceOptions]rlposden=DEFAULT
[DLOptions]simplifyTimeout=0
[Strategy]ActiveStrategy=DLStrategy
[DLOptions]usePowersetIterativeReduce=true
[DLOptions]quadricTimeoutIncreaseFactor=0
[DLOptions]FOStrategy=LAZY
[ReduceOptions]rlqepnf=DEFAULT
[DLOptions]applyLocalReduce=OFF
[DLOptions]quantifierEliminator=Mathematica
"
}

/* REQUIRED INFORMATION:

Statistics can be obtained from Proof -> Show Proof Statistics.

Name: Milda Zizyte & Felix Hutchison
AndrewID: mzizyte; fhutchin
KeYmaera version: 3.6.12
Backends used (Mathematica, Z3, ...): Mathematica
Nodes: 
Branches: 
Interactive steps: 
Arithmetic Solver: 
Time: 
Proof completes (Y/N): 

*/

\functions {
R Width;
R Height;
R vel;
}

\programVariables {
R bx;
R by;
R vx;
R vy;

R paddleWidth;
//R Px;
R Py;
R Pvy;
R Pvmax;
//R Ox;
R Oy;
}

\problem {
   (  (  (  (  (  (  (  (  (  (  (  (  (  (   (Width)
                                           >  (0))
                                        & (   (Height)
                                           >  (0)))
                                     & ((bx) = (Width)))
                                  & ((by) >  (0)))
                               & ((vx) <  (0)))
                            & ((by) <  (Height)))
                         & (  (  ((vx) ^ (2))
                               + ((vy) ^ (2)))
                            = ((vel) ^ (2))))
                      & ((vel) >  (0)))
                   & ((paddleWidth) >  (0)))
                & (((Pvmax) ^ (2)) >  ((vy) ^ (2))))
             & ((Pvmax) >  (0)))
          & (   ((-(Height)) * ((Width) / (vel)))
             <= (Pvmax)))
       & ((Py) <  (Height)))
    & ((Py) >  (0)))
-> (\<
      ((if(((by) = (0)) & ((vy) < (0)))
          then
            vy := (-(vy))
          else
            if(((by) = (Height)) & ((vy) > (0)))
            then
              vy := (-(vy))
            fi
          fi) ;
        ((if((by) > (Py))
            then
              Pvy := (Pvmax)
            else
              Pvy := (-(Pvmax))
            fi) ;
          ((Oy := (by)) ;
            ({(bx') = (vx), 
              (by') = (vy), 
              (Py') = (Pvy), 
              (bx) >= (0), 
              (bx) <= (Width), 
              (by) >= (0), 
              (by) <= (Height)}))))*
      \> (  (  ((bx) >= (0))
             & ((by) >= ((Py) - (paddleWidth))))
          & ((by) <= ((Py) + (paddleWidth)))))

}

\proof {
(keyLog "0" (keyUser "Cas" ) (keyVersion "b3a7f5979673d197848e89aa57b0943d144cf2f0"))

(autoModeTime "53908")

(nodes "1663")

(branches "345")

(interactiveSteps "5")

(branch "dummy ID"
(rule "imp_right" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3690657"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3690686"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3690725"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3690759"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3690802"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3690841"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3690880"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3690919"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3690958"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3690993"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3691027"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3691061"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3691100"))
(rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3691135"))
(rule "loop_unwind_dia" (formula "15") (userinteraction) (quantifierEliminator "Mathematica") (nodenum "3691437"))
(rule "or_right" (formula "15") (userinteraction) (quantifierEliminator "Mathematica") (nodenum "3691580"))
(rule "modality_split_right" (formula "16") (userinteraction) (quantifierEliminator "Mathematica") (nodenum "3691648"))
(rule "if_then_else_right" (formula "16") (userinteraction) (quantifierEliminator "Mathematica") (nodenum "3691696"))
(branch " by = 0 & vy < 0"
   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3742657"))
   (rule "close_ineq_eq_2" (formula "1") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3742700"))
)
(branch " ! by = 0 & vy < 0"
   (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3693478"))
   (rule "and_right" (formula "16") (quantifierEliminator "Mathematica") (nodenum "3693497"))
   (branch " Case 1"
      (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3693537"))
      (branch " Case 1"
         (rule "and_right" (formula "16") (quantifierEliminator "Mathematica") (nodenum "3693580"))
         (branch " Case 1"
            (rule "if_then_right" (formula "17") (quantifierEliminator "Mathematica") (nodenum "3693632"))
            (branch " by = Height & vy > 0"
               (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3693712"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3693768"))
               (rule "applyEq" (formula "17") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3693793"))
               (rule "applyEq" (formula "6") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3693814"))
               (rule "applyEq" (formula "7") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3693817"))
               (rule "applyEq" (formula "17") (term "0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3693839"))
               (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3693863"))
               (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3693883"))
               (branch " by > Py"
                  (builtin "Update Simplification" (formula "1") (nodenum "3693986"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3694014"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3694040"))
                  (rule "applyEq" (formula "1") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3694058"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3694085"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3694106"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3694134"))
                  (rule "applyEq" (formula "19") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3694157"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3694180"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3694468"))
                  (rule "ex_right" (formula "19") (newnames "T_0_0") (quantifierEliminator "Mathematica") (nodenum "3694690"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3695000"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3695054"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3695056"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_1") (quantifierEliminator "Mathematica") (nodenum "3695091"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3695232"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3695269"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3695317"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3695363"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3695411"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3695478"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3695513"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3695529"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3695556"))
                              )
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3695571"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3695606"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3695622"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3695649"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3695665"))
                        (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3695689"))
                        (opengoal " Height >  Py,  by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  Height = 0,  Width >= 0,  {Oy:=Height ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_0_0 ||    bx:=Width + T_0_0 * vx ||    by:=Height + T_0_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
               (branch " ! by > Py"
                  (builtin "Update Simplification" (formula "1") (nodenum "3695714"))
                  (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3695741"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3695766"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3695798"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3695823"))
                  (rule "applyEq" (formula "16") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3695839"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3695863"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3695891"))
                  (rule "applyEq" (formula "19") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3695915"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3695938"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3696226"))
                  (rule "ex_right" (formula "19") (newnames "T_2_0") (quantifierEliminator "Mathematica") (nodenum "3696454"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3696764"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3696818"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3696820"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_4") (quantifierEliminator "Mathematica") (nodenum "3696855"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3696996"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3697033"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3697081"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3697127"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3697175"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3697242"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3697277"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3697293"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3697320"))
                              )
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3697335"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3697370"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3697386"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3697413"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3697429"))
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3697452"))
                        (opengoal " by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  Height >  Py,  Height = 0,  Width >= 0,  {Oy:=Height ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_2_0 ||    bx:=Width + T_2_0 * vx ||    by:=Height + T_2_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
            )
            (branch " ! by = Height & vy > 0"
               (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3697479"))
               (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3697498"))
               (branch " Case 1"
                  (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3697542"))
                  (rule "applyEq" (formula "17") (term "0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3697556"))
                  (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3697565"))
                  (branch " by > Py"
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3697603"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3697621"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3697637"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3697665"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3697687"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3697975"))
                     (rule "ex_right" (formula "19") (newnames "T_4_0") (quantifierEliminator "Mathematica") (nodenum "3698179"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3698483"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3698536"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3698538"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_7") (quantifierEliminator "Mathematica") (nodenum "3698573"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3698714"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3698751"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3698799"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3698845"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3698893"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3698956"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3698991"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3742712"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3742739"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3699000"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3699009"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3699020"))
                           (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by = Height,  by = 0,  Width >= 0,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_4_0 ||    bx:=Width + T_4_0 * vx ||    by:=by + T_4_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
                  (branch " ! by > Py"
                     (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3699046"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3699055"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3699073"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3699089"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3699117"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3699139"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3699427"))
                     (rule "ex_right" (formula "19") (newnames "T_6_0") (quantifierEliminator "Mathematica") (nodenum "3699637"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3699941"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3699994"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3699996"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_10") (quantifierEliminator "Mathematica") (nodenum "3700031"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3700172"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3700209"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3700257"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3700303"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3700351"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3700414"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3700449"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3700461"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3700488"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3700497"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3700506"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3700517"))
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  by = Height,  by = 0,  Width >= 0,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_6_0 ||    bx:=Width + T_6_0 * vx ||    by:=by + T_6_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
               )
               (branch " Case 2"
                  (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3700544"))
                  (rule "applyEq" (formula "17") (term "0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3700553"))
                  (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3700562"))
                  (branch " by > Py"
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3700590"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3700598"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3700606"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3700614"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3700624"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3700912"))
                     (rule "ex_right" (formula "19") (newnames "T_8_0") (quantifierEliminator "Mathematica") (nodenum "3701116"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3701420"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3701473"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3701475"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_13") (quantifierEliminator "Mathematica") (nodenum "3701510"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3701651"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3701688"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3701736"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3701782"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3701830"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3701893"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3701928"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3701940"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3701967"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3701976"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3701985"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3701996"))
                           (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy >  0,  by = 0,  Width >= 0,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_8_0 ||    bx:=Width + T_8_0 * vx ||    by:=by + T_8_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
                  (branch " ! by > Py"
                     (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3702022"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3702031"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3702039"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3702047"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3702055"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3702065"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3702353"))
                     (rule "ex_right" (formula "19") (newnames "T_10_0") (quantifierEliminator "Mathematica") (nodenum "3702563"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3702867"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3702920"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3702922"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_16") (quantifierEliminator "Mathematica") (nodenum "3702957"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3703098"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3703135"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3703183"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3703229"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3703277"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3703340"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3703375"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3703387"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3703414"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3703423"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3703432"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3703443"))
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  by = 0,  Width >= 0,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_10_0 ||    bx:=Width + T_10_0 * vx ||    by:=by + T_10_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
               )
            )
         )
         (branch " Case 2"
            (rule "if_then_right" (formula "17") (quantifierEliminator "Mathematica") (nodenum "3703475"))
            (branch " by = Height & vy > 0"
               (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3703503"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3703526"))
               (rule "applyEq" (formula "18") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3703534"))
               (rule "applyEq" (formula "17") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3703558"))
               (rule "applyEq" (formula "6") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3703566"))
               (rule "applyEq" (formula "7") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3703569"))
               (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3703578"))
               (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3703588"))
               (branch " by > Py"
                  (builtin "Update Simplification" (formula "1") (nodenum "3703615"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3703624"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3703632"))
                  (rule "applyEq" (formula "1") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3703640"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3703648"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3703657"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3703665"))
                  (rule "applyEq" (formula "19") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3703676"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3703687"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3703975"))
                  (rule "ex_right" (formula "19") (newnames "T_12_0") (quantifierEliminator "Mathematica") (nodenum "3742741"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3743051"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3743105"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3743107"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_90") (quantifierEliminator "Mathematica") (nodenum "3743142"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3743283"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3743320"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3743368"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3743414"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3743462"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3743529"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3743564"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3743580"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3743607"))
                              )
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3743622"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3743657"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3743673"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3743700"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3743716"))
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3743740"))
                        (opengoal " Height >  Py,  by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  Height = 0,  Height >= Py - paddleWidth,  {Oy:=Height ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_12_0 ||    bx:=Width + T_12_0 * vx ||    by:=Height + T_12_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
               (branch " ! by > Py"
                  (builtin "Update Simplification" (formula "1") (nodenum "3704196"))
                  (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3704204"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3704213"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3704221"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3704230"))
                  (rule "applyEq" (formula "16") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3704238"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3704247"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3704255"))
                  (rule "applyEq" (formula "19") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3704266"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3704277"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3704565"))
                  (rule "ex_right" (formula "19") (newnames "T_13_0") (quantifierEliminator "Mathematica") (nodenum "3704793"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3705103"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3705157"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3705159"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_20") (quantifierEliminator "Mathematica") (nodenum "3705194"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3705335"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3705372"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3705420"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3705466"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3705514"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3705581"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3705616"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3705632"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3705659"))
                              )
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3705674"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3705709"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3705725"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3705752"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3705768"))
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3705791"))
                        (opengoal " by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  Height >  Py,  Height = 0,  Height >= Py - paddleWidth,  {Oy:=Height ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_13_0 ||    bx:=Width + T_13_0 * vx ||    by:=Height + T_13_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
            )
            (branch " ! by = Height & vy > 0"
               (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3705818"))
               (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3705827"))
               (branch " Case 1"
                  (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3705841"))
                  (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3705850"))
                  (branch " by > Py"
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3705878"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3705886"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3705894"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3705902"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3705912"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3706200"))
                     (rule "ex_right" (formula "19") (newnames "T_15_0") (quantifierEliminator "Mathematica") (nodenum "3706404"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3706708"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3706761"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3706763"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_23") (quantifierEliminator "Mathematica") (nodenum "3706798"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3706939"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3706976"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3707024"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3707070"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3707118"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3707181"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3707216"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3707228"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3707255"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3707264"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3707273"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3707284"))
                           (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by = Height,  by = 0,  by >= Py - paddleWidth,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_15_0 ||    bx:=Width + T_15_0 * vx ||    by:=by + T_15_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
                  (branch " ! by > Py"
                     (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3707310"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3707319"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3707327"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3707335"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3707343"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3707353"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3707641"))
                     (rule "ex_right" (formula "19") (newnames "T_17_0") (quantifierEliminator "Mathematica") (nodenum "3707851"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3708155"))
                     (branch " Case 1"
                        (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  by = Height,  by = 0,  by >= Py - paddleWidth,  T_17_0 >= 0   ")
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3708207"))
                        (branch " Case 1"
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  by = Height,  by = 0,  by >= Py - paddleWidth,  \\forall R ts_25.     (   0 <= ts_25 & ts_25 <= T_17_0     ->       bx + ts_25 * vx >= 0            & bx + ts_25 * vx <= Width          & by + ts_25 * vy >= 0        & by + ts_25 * vy <= Height)   ")
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3708247"))
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  by = Height,  by = 0,  by >= Py - paddleWidth,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_17_0 ||    bx:=Width + T_17_0 * vx ||    by:=by + T_17_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
               )
               (branch " Case 2"
                  (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3708273"))
                  (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3708282"))
                  (branch " by > Py"
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3708310"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3708318"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3708326"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3708334"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3708344"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3708632"))
                     (rule "ex_right" (formula "19") (newnames "T_19_0") (quantifierEliminator "Mathematica") (nodenum "3708836"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3709140"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3709193"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3709195"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_28") (quantifierEliminator "Mathematica") (nodenum "3709230"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3709371"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3709408"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3709456"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3709502"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3709550"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3709613"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3709648"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3709660"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3709687"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3709696"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3709705"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3709716"))
                           (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy >  0,  by = 0,  by >= Py - paddleWidth,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_19_0 ||    bx:=Width + T_19_0 * vx ||    by:=by + T_19_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
                  (branch " ! by > Py"
                     (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3743766"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3743775"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3743783"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3743791"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3743799"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3743809"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3744097"))
                     (rule "ex_right" (formula "19") (newnames "T_64_0") (quantifierEliminator "Mathematica") (nodenum "3744307"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3744611"))
                     (branch " Case 1"
                        (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  by = 0,  by >= Py - paddleWidth,  T_64_0 >= 0   ")
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3744663"))
                        (branch " Case 1"
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  by = 0,  by >= Py - paddleWidth,  \\forall R ts_91.     (   0 <= ts_91 & ts_91 <= T_64_0     ->       bx + ts_91 * vx >= 0            & bx + ts_91 * vx <= Width          & by + ts_91 * vy >= 0        & by + ts_91 * vy <= Height)   ")
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3744703"))
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  by = 0,  by >= Py - paddleWidth,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_64_0 ||    bx:=Width + T_64_0 * vx ||    by:=by + T_64_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
               )
            )
         )
      )
      (branch " Case 2"
         (rule "and_right" (formula "16") (quantifierEliminator "Mathematica") (nodenum "3709742"))
         (branch " Case 1"
            (rule "if_then_right" (formula "17") (quantifierEliminator "Mathematica") (nodenum "3709762"))
            (branch " by = Height & vy > 0"
               (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3709790"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3709810"))
               (rule "applyEq" (formula "18") (term "0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3709820"))
               (rule "applyEq" (formula "8") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3709828"))
               (rule "applyEq" (formula "6") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3709836"))
               (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3709840"))
               (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3709849"))
               (branch " by > Py"
                  (builtin "Update Simplification" (formula "1") (nodenum "3709876"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3709884"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3709892"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3709901"))
                  (rule "applyEq" (formula "1") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3709909"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3709917"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3709925"))
                  (rule "applyEq" (formula "19") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3709937"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3709948"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3710236"))
                  (rule "ex_right" (formula "19") (newnames "T_21_0") (quantifierEliminator "Mathematica") (nodenum "3710458"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3710768"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3710822"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3710824"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_31") (quantifierEliminator "Mathematica") (nodenum "3710859"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3711000"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3711037"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3711085"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3711131"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3711179"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3711246"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3711281"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3711297"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3711324"))
                              )
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3711339"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3711374"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3711390"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3711417"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3711433"))
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3711457"))
                        (opengoal " Height >  Py,  by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy <  0,  Width >= 0,  {Oy:=Height ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_21_0 ||    bx:=Width + T_21_0 * vx ||    by:=Height + T_21_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
               (branch " ! by > Py"
                  (builtin "Update Simplification" (formula "1") (nodenum "3711482"))
                  (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3711490"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3711499"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3711507"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3711516"))
                  (rule "applyEq" (formula "16") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3711524"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3711533"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3711541"))
                  (rule "applyEq" (formula "19") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3711552"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3711563"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3711851"))
                  (rule "ex_right" (formula "19") (newnames "T_23_0") (quantifierEliminator "Mathematica") (nodenum "3712079"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3712389"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3712443"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3712445"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_34") (quantifierEliminator "Mathematica") (nodenum "3712480"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3712621"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3712658"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3712706"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3712752"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3712800"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3712867"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3712902"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3712918"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3712945"))
                              )
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3712960"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3712995"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3713011"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3713038"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3713054"))
                        (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3713078"))
                        (opengoal " by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  Height >  Py,  vy <  0,  Width >= 0,  {Oy:=Height ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_23_0 ||    bx:=Width + T_23_0 * vx ||    by:=Height + T_23_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
            )
            (branch " ! by = Height & vy > 0"
               (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3713104"))
               (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3713113"))
               (branch " Case 1"
                  (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3713128"))
                  (rule "applyEq" (formula "17") (term "0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3713137"))
                  (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3713146"))
                  (branch " by > Py"
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3713174"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3713182"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3713190"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3713198"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3713208"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3713496"))
                     (rule "ex_right" (formula "19") (newnames "T_25_0") (quantifierEliminator "Mathematica") (nodenum "3713700"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3714004"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3714057"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3714059"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_37") (quantifierEliminator "Mathematica") (nodenum "3714094"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3714235"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3714272"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3714320"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3714366"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3714414"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3714477"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3714512"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3714524"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3714551"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3714560"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3714569"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3714580"))
                           (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by = Height,  vy <  0,  Width >= 0,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_25_0 ||    bx:=Width + T_25_0 * vx ||    by:=by + T_25_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
                  (branch " ! by > Py"
                     (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3714603"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3714612"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3714620"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3714628"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3714636"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3714646"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3714934"))
                     (rule "ex_right" (formula "19") (newnames "T_27_0") (quantifierEliminator "Mathematica") (nodenum "3715144"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3715448"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3715501"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3715503"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_40") (quantifierEliminator "Mathematica") (nodenum "3715538"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3715679"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3715716"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3715764"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3715810"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3715858"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3715921"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3715956"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3715968"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3715995"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3716004"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3716013"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3716024"))
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  by = Height,  vy <  0,  Width >= 0,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_27_0 ||    bx:=Width + T_27_0 * vx ||    by:=by + T_27_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
               )
               (branch " Case 2"
                  (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3716051"))
                  (rule "applyEq" (formula "17") (term "0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3716060"))
                  (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3716069"))
                  (branch " by > Py"
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3716097"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3716105"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3716113"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3716121"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3716131"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3716419"))
                     (rule "ex_right" (formula "19") (newnames "T_29_0") (quantifierEliminator "Mathematica") (nodenum "3716623"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3716927"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3716980"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3716982"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_43") (quantifierEliminator "Mathematica") (nodenum "3717017"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3717158"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3717195"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3717243"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3717289"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3717337"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3717400"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3717435"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3717447"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3717474"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3717483"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3717492"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3717503"))
                           (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy >  0,  vy <  0,  Width >= 0,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_29_0 ||    bx:=Width + T_29_0 * vx ||    by:=by + T_29_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
                  (branch " ! by > Py"
                     (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3717529"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3717538"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3717546"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3717554"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3717562"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3717572"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3717860"))
                     (rule "ex_right" (formula "19") (newnames "T_32_0") (quantifierEliminator "Mathematica") (nodenum "3718070"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3718374"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3718427"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3718429"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_47") (quantifierEliminator "Mathematica") (nodenum "3718464"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3718605"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3718642"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3718690"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3718736"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3718784"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3718847"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3718882"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3718894"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3718921"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3718930"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3718939"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3718950"))
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  vy <  0,  Width >= 0,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_32_0 ||    bx:=Width + T_32_0 * vx ||    by:=by + T_32_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
               )
            )
         )
         (branch " Case 2"
            (rule "if_then_right" (formula "17") (quantifierEliminator "Mathematica") (nodenum "3718982"))
            (branch " by = Height & vy > 0"
               (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3719010"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3719031"))
               (rule "applyEq" (formula "18") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3719040"))
               (rule "applyEq" (formula "6") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3719048"))
               (rule "applyEq" (formula "7") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3719051"))
               (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3719060"))
               (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3719069"))
               (branch " by > Py"
                  (builtin "Update Simplification" (formula "1") (nodenum "3719096"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3719104"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3719112"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3719121"))
                  (rule "applyEq" (formula "1") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3719129"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3719137"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3719145"))
                  (rule "applyEq" (formula "19") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3719157"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3719168"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3719456"))
                  (rule "ex_right" (formula "19") (newnames "T_34_0") (quantifierEliminator "Mathematica") (nodenum "3719678"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3719988"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3720042"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3720044"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_50") (quantifierEliminator "Mathematica") (nodenum "3720079"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3744732"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3744769"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3744817"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3744863"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3744911"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3744978"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3745013"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3745029"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3745056"))
                              )
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3745071"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3745106"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3745122"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3745149"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3720234"))
                        (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3720257"))
                        (opengoal " Height >  Py,  by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy <  0,  Height >= Py - paddleWidth,  {Oy:=Height ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_34_0 ||    bx:=Width + T_34_0 * vx ||    by:=Height + T_34_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
               (branch " ! by > Py"
                  (builtin "Update Simplification" (formula "1") (nodenum "3720283"))
                  (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3720291"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3720300"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3720308"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3720317"))
                  (rule "applyEq" (formula "16") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3720325"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3720333"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3720341"))
                  (rule "applyEq" (formula "19") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3720353"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3720364"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3720652"))
                  (rule "ex_right" (formula "19") (newnames "T_36_0") (quantifierEliminator "Mathematica") (nodenum "3720880"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3721190"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3721244"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3721246"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_53") (quantifierEliminator "Mathematica") (nodenum "3721281"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3721422"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3721459"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3721507"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3721553"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3721601"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3721668"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3721703"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3721719"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3721746"))
                              )
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3721761"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3721796"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3721812"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3721839"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3721855"))
                        (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3721879"))
                        (opengoal " by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  Height >  Py,  vy <  0,  Height >= Py - paddleWidth,  {Oy:=Height ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_36_0 ||    bx:=Width + T_36_0 * vx ||    by:=Height + T_36_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
            )
            (branch " ! by = Height & vy > 0"
               (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3721905"))
               (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3721914"))
               (branch " Case 1"
                  (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3721928"))
                  (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3721937"))
                  (branch " by > Py"
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3721965"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3721973"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3721981"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3721989"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3721999"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3722287"))
                     (rule "ex_right" (formula "19") (newnames "T_38_0") (quantifierEliminator "Mathematica") (nodenum "3722491"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3722795"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3722848"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3722850"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_56") (quantifierEliminator "Mathematica") (nodenum "3722885"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3723026"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3723063"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3723111"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3723157"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3723205"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3723268"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3723303"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3723315"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3723342"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3723351"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3723360"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3723371"))
                           (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by = Height,  vy <  0,  by >= Py - paddleWidth,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_38_0 ||    bx:=Width + T_38_0 * vx ||    by:=by + T_38_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
                  (branch " ! by > Py"
                     (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3723397"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3723406"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3723414"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3723422"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3723430"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3723440"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3723728"))
                     (rule "ex_right" (formula "19") (newnames "T_40_0") (quantifierEliminator "Mathematica") (nodenum "3723938"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3724242"))
                     (branch " Case 1"
                        (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  by = Height,  vy <  0,  by >= Py - paddleWidth,  T_40_0 >= 0   ")
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3724294"))
                        (branch " Case 1"
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  by = Height,  vy <  0,  by >= Py - paddleWidth,  \\forall R ts_58.     (   0 <= ts_58 & ts_58 <= T_40_0     ->       bx + ts_58 * vx >= 0            & bx + ts_58 * vx <= Width          & by + ts_58 * vy >= 0        & by + ts_58 * vy <= Height)   ")
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3724334"))
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  by = Height,  vy <  0,  by >= Py - paddleWidth,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_40_0 ||    bx:=Width + T_40_0 * vx ||    by:=by + T_40_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
               )
               (branch " Case 2"
                  (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3724360"))
                  (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3724369"))
                  (branch " by > Py"
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3724397"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3724405"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3724413"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3724421"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3724431"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3724719"))
                     (rule "ex_right" (formula "19") (newnames "T_42_0") (quantifierEliminator "Mathematica") (nodenum "3724923"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3725227"))
                     (branch " Case 1"
                        (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3725280"))
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3725282"))
                        (branch " Case 1"
                           (rule "all_right" (formula "19") (inst "skC=ts_61") (quantifierEliminator "Mathematica") (nodenum "3725317"))
                           (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3725458"))
                           (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3725495"))
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3725543"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3725589"))
                              (branch " Case 1"
                                 (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3725637"))
                                 (branch " Case 1"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3725700"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3725735"))
                                 )
                                 (branch " Case 2"
                                    (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3725747"))
                                    (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3725774"))
                                 )
                              )
                              (branch " Case 2"
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3725783"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3745158"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3725794"))
                           (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy >  0,  vy <  0,  by >= Py - paddleWidth,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_42_0 ||    bx:=Width + T_42_0 * vx ||    by:=by + T_42_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
                  (branch " ! by > Py"
                     (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3725820"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3725829"))
                     (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3725837"))
                     (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3725845"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3725853"))
                     (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3725863"))
                     (builtin "Update Simplification" (formula "19") (nodenum "3726151"))
                     (rule "ex_right" (formula "19") (newnames "T_44_0") (quantifierEliminator "Mathematica") (nodenum "3726361"))
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3726665"))
                     (branch " Case 1"
                        (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  vy <  0,  by >= Py - paddleWidth,  T_44_0 >= 0   ")
                     )
                     (branch " Case 2"
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3726717"))
                        (branch " Case 1"
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  vy <  0,  by >= Py - paddleWidth,  \\forall R ts_63.     (   0 <= ts_63 & ts_63 <= T_44_0     ->       bx + ts_63 * vx >= 0            & bx + ts_63 * vx <= Width          & by + ts_63 * vy >= 0        & by + ts_63 * vy <= Height)   ")
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3726757"))
                           (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  vy <  0,  by >= Py - paddleWidth,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_44_0 ||    bx:=Width + T_44_0 * vx ||    by:=by + T_44_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                        )
                     )
                  )
               )
            )
         )
      )
   )
   (branch " Case 2"
      (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3726783"))
      (branch " Case 1"
         (rule "if_then_right" (formula "17") (quantifierEliminator "Mathematica") (nodenum "3726803"))
         (branch " by = Height & vy > 0"
            (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3726831"))
            (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3726853"))
            (rule "applyEq" (formula "18") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3726862"))
            (rule "applyEq" (formula "8") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3726886"))
            (rule "applyEq" (formula "6") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3726894"))
            (rule "applyEq" (formula "16") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3726897"))
            (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3726905"))
            (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3726916"))
            (branch " by > Py"
               (builtin "Update Simplification" (formula "1") (nodenum "3726943"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3726952"))
               (builtin "Update Simplification" (formula "19") (nodenum "3726960"))
               (rule "applyEq" (formula "1") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3726968"))
               (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3726976"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3726985"))
               (builtin "Update Simplification" (formula "19") (nodenum "3726993"))
               (rule "applyEq" (formula "19") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3727004"))
               (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3727015"))
               (builtin "Update Simplification" (formula "19") (nodenum "3727303"))
               (rule "ex_right" (formula "19") (newnames "T_46_0") (quantifierEliminator "Mathematica") (nodenum "3727525"))
               (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3727835"))
               (branch " Case 1"
                  (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3727889"))
               )
               (branch " Case 2"
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3727891"))
                  (branch " Case 1"
                     (rule "all_right" (formula "19") (inst "skC=ts_66") (quantifierEliminator "Mathematica") (nodenum "3727926"))
                     (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3728067"))
                     (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3728104"))
                     (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3728152"))
                     (branch " Case 1"
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3728198"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3728246"))
                           (branch " Case 1"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3728313"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3728348"))
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3728364"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3728391"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3728406"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3728441"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3728457"))
                        (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3728484"))
                     )
                  )
                  (branch " Case 2"
                     (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3728500"))
                     (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3728524"))
                     (opengoal " Height >  Py,  by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  Height = 0,  Height <= Py + paddleWidth,  {Oy:=Height ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_46_0 ||    bx:=Width + T_46_0 * vx ||    by:=Height + T_46_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                  )
               )
            )
            (branch " ! by > Py"
               (builtin "Update Simplification" (formula "1") (nodenum "3728549"))
               (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3728557"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3728566"))
               (builtin "Update Simplification" (formula "19") (nodenum "3728574"))
               (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3728583"))
               (rule "applyEq" (formula "16") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3728591"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3728600"))
               (builtin "Update Simplification" (formula "19") (nodenum "3728608"))
               (rule "applyEq" (formula "19") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3728619"))
               (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3728630"))
               (builtin "Update Simplification" (formula "19") (nodenum "3728918"))
               (rule "ex_right" (formula "19") (newnames "T_47_0") (quantifierEliminator "Mathematica") (nodenum "3729146"))
               (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3729456"))
               (branch " Case 1"
                  (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3729510"))
               )
               (branch " Case 2"
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3729512"))
                  (branch " Case 1"
                     (rule "all_right" (formula "19") (inst "skC=ts_68") (quantifierEliminator "Mathematica") (nodenum "3729547"))
                     (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3729688"))
                     (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3729725"))
                     (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3729773"))
                     (branch " Case 1"
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3729819"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3729867"))
                           (branch " Case 1"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3729934"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3729969"))
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3729985"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3730012"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3730027"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3730062"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3730078"))
                        (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3730105"))
                     )
                  )
                  (branch " Case 2"
                     (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3730121"))
                     (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3730144"))
                     (opengoal " by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  Height >  Py,  Height = 0,  Height <= Py + paddleWidth,  {Oy:=Height ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_47_0 ||    bx:=Width + T_47_0 * vx ||    by:=Height + T_47_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                  )
               )
            )
         )
         (branch " ! by = Height & vy > 0"
            (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3730171"))
            (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3730180"))
            (branch " Case 1"
               (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3730194"))
               (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3730203"))
               (branch " by > Py"
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3730231"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3730239"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3730247"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3730255"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3730265"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3730553"))
                  (rule "ex_right" (formula "19") (newnames "T_49_0") (quantifierEliminator "Mathematica") (nodenum "3730757"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3745160"))
                  (branch " Case 1"
                     (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by = Height,  by = 0,  by <= Py + paddleWidth,  T_49_0 >= 0   ")
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3745212"))
                     (branch " Case 1"
                        (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by = Height,  by = 0,  by <= Py + paddleWidth,  \\forall R ts_70.     (   0 <= ts_70 & ts_70 <= T_49_0     ->       bx + ts_70 * vx >= 0            & bx + ts_70 * vx <= Width          & by + ts_70 * vy >= 0        & by + ts_70 * vy <= Height)   ")
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3745252"))
                        (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by = Height,  by = 0,  by <= Py + paddleWidth,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_49_0 ||    bx:=Width + T_49_0 * vx ||    by:=by + T_49_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
               (branch " ! by > Py"
                  (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3731061"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3731070"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3731078"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3731086"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3731094"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3731104"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3731392"))
                  (rule "ex_right" (formula "19") (newnames "T_51_0") (quantifierEliminator "Mathematica") (nodenum "3731602"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3731906"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3731959"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3731961"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_73") (quantifierEliminator "Mathematica") (nodenum "3731996"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3732137"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3732174"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3732222"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3732268"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3732316"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3732379"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3732414"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3732426"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3732453"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3732462"))
                           )
                        )
                        (branch " Case 2"
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3732471"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3732482"))
                        (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  by = Height,  by = 0,  by <= Py + paddleWidth,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_51_0 ||    bx:=Width + T_51_0 * vx ||    by:=by + T_51_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
            )
            (branch " Case 2"
               (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3732508"))
               (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3732517"))
               (branch " by > Py"
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3732545"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3732553"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3732561"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3732569"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3732579"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3732867"))
                  (rule "ex_right" (formula "19") (newnames "T_53_0") (quantifierEliminator "Mathematica") (nodenum "3733071"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3733375"))
                  (branch " Case 1"
                     (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy >  0,  by = 0,  by <= Py + paddleWidth,  T_53_0 >= 0   ")
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3733427"))
                     (branch " Case 1"
                        (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy >  0,  by = 0,  by <= Py + paddleWidth,  \\forall R ts_75.     (   0 <= ts_75 & ts_75 <= T_53_0     ->       bx + ts_75 * vx >= 0            & bx + ts_75 * vx <= Width          & by + ts_75 * vy >= 0        & by + ts_75 * vy <= Height)   ")
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3733467"))
                        (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy >  0,  by = 0,  by <= Py + paddleWidth,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_53_0 ||    bx:=Width + T_53_0 * vx ||    by:=by + T_53_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
               (branch " ! by > Py"
                  (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3733493"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3733502"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3733510"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3733518"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3733526"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3733536"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3733824"))
                  (rule "ex_right" (formula "19") (newnames "T_54_0") (quantifierEliminator "Mathematica") (nodenum "3734034"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3734338"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3734391"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3734393"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_77") (quantifierEliminator "Mathematica") (nodenum "3734428"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3734569"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3734606"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3734654"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3734700"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3734748"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3734811"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3734846"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3734858"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3734885"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3734894"))
                           )
                        )
                        (branch " Case 2"
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3734903"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3734914"))
                        (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  by = 0,  by <= Py + paddleWidth,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_54_0 ||    bx:=Width + T_54_0 * vx ||    by:=by + T_54_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
            )
         )
      )
      (branch " Case 2"
         (rule "if_then_right" (formula "17") (quantifierEliminator "Mathematica") (nodenum "3734946"))
         (branch " by = Height & vy > 0"
            (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3734974"))
            (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3734995"))
            (rule "applyEq" (formula "18") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3735004"))
            (rule "applyEq" (formula "6") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3735012"))
            (rule "applyEq" (formula "7") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3735015"))
            (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3735024"))
            (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3735033"))
            (branch " by > Py"
               (builtin "Update Simplification" (formula "1") (nodenum "3735060"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3735068"))
               (builtin "Update Simplification" (formula "19") (nodenum "3735076"))
               (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3735085"))
               (rule "applyEq" (formula "1") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3735093"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3735101"))
               (builtin "Update Simplification" (formula "19") (nodenum "3735109"))
               (rule "applyEq" (formula "19") (term "0") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3735121"))
               (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3735132"))
               (builtin "Update Simplification" (formula "19") (nodenum "3735420"))
               (rule "ex_right" (formula "19") (newnames "T_56_0") (quantifierEliminator "Mathematica") (nodenum "3735642"))
               (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3735952"))
               (branch " Case 1"
                  (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3736006"))
               )
               (branch " Case 2"
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3736008"))
                  (branch " Case 1"
                     (rule "all_right" (formula "19") (inst "skC=ts_80") (quantifierEliminator "Mathematica") (nodenum "3736043"))
                     (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3736184"))
                     (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3736221"))
                     (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3736269"))
                     (branch " Case 1"
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3736315"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3736363"))
                           (branch " Case 1"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3736430"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3736465"))
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "8") (quantifierEliminator "Mathematica") (nodenum "3736481"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3736508"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3736523"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3736558"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3736574"))
                        (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3745278"))
                     )
                  )
                  (branch " Case 2"
                     (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "6") (quantifierEliminator "Mathematica") (nodenum "3736615"))
                     (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3736638"))
                     (opengoal " Height >  Py,  by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy <  0,  Height <= Py + paddleWidth,  {Oy:=Height ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_56_0 ||    bx:=Width + T_56_0 * vx ||    by:=Height + T_56_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                  )
               )
            )
            (branch " ! by > Py"
               (builtin "Update Simplification" (formula "1") (nodenum "3736664"))
               (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3736672"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3736681"))
               (builtin "Update Simplification" (formula "19") (nodenum "3736689"))
               (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3736698"))
               (rule "applyEq" (formula "16") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3736706"))
               (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3736714"))
               (builtin "Update Simplification" (formula "19") (nodenum "3736722"))
               (rule "applyEq" (formula "19") (term "0") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3736734"))
               (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3736745"))
               (builtin "Update Simplification" (formula "19") (nodenum "3737033"))
               (rule "ex_right" (formula "19") (newnames "T_58_0") (quantifierEliminator "Mathematica") (nodenum "3737261"))
               (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3737571"))
               (branch " Case 1"
                  (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3737625"))
               )
               (branch " Case 2"
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3737627"))
                  (branch " Case 1"
                     (rule "all_right" (formula "19") (inst "skC=ts_83") (quantifierEliminator "Mathematica") (nodenum "3737662"))
                     (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3737803"))
                     (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3737840"))
                     (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3737888"))
                     (branch " Case 1"
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3737934"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3737982"))
                           (branch " Case 1"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3738049"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3738084"))
                           )
                           (branch " Case 2"
                              (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "7") (quantifierEliminator "Mathematica") (nodenum "3738100"))
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3738127"))
                           )
                        )
                        (branch " Case 2"
                           (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3738142"))
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3738177"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3738193"))
                        (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3738220"))
                     )
                  )
                  (branch " Case 2"
                     (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3738236"))
                     (rule "applyEq" (formula "19") (term "0,4") (ifseqformula "1") (quantifierEliminator "Mathematica") (nodenum "3738260"))
                     (opengoal " by = Height,  vy >  0,  Width >  0,  Height >  0,  bx = Width,  vx <  0,  Height <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  Height >  Py,  vy <  0,  Height <= Py + paddleWidth,  {Oy:=Height ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_58_0 ||    bx:=Width + T_58_0 * vx ||    by:=Height + T_58_0 * -vy ||    vy:=-vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                  )
               )
            )
         )
         (branch " ! by = Height & vy > 0"
            (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3738286"))
            (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3738295"))
            (branch " Case 1"
               (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3738309"))
               (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3738318"))
               (branch " by > Py"
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3738346"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3738354"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3738362"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3738370"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3738380"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3738668"))
                  (rule "ex_right" (formula "19") (newnames "T_60_0") (quantifierEliminator "Mathematica") (nodenum "3738872"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3739176"))
                  (branch " Case 1"
                     (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by = Height,  vy <  0,  by <= Py + paddleWidth,  T_60_0 >= 0   ")
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3739228"))
                     (branch " Case 1"
                        (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by = Height,  vy <  0,  by <= Py + paddleWidth,  \\forall R ts_85.     (   0 <= ts_85 & ts_85 <= T_60_0     ->       bx + ts_85 * vx >= 0            & bx + ts_85 * vx <= Width          & by + ts_85 * vy >= 0        & by + ts_85 * vy <= Height)   ")
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3739268"))
                        (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by = Height,  vy <  0,  by <= Py + paddleWidth,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_60_0 ||    bx:=Width + T_60_0 * vx ||    by:=by + T_60_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
               (branch " ! by > Py"
                  (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3739294"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3739303"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3739311"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3739319"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3739327"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3739337"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3739625"))
                  (rule "ex_right" (formula "19") (newnames "T_61_0") (quantifierEliminator "Mathematica") (nodenum "3739835"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3740139"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3740192"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3740194"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_87") (quantifierEliminator "Mathematica") (nodenum "3740229"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3740370"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3740407"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3740455"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3740501"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3740549"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3740612"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3740647"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3740659"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3740686"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3740695"))
                           )
                        )
                        (branch " Case 2"
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3740704"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3740715"))
                        (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  by = Height,  vy <  0,  by <= Py + paddleWidth,  {Oy:=by ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_61_0 ||    bx:=Width + T_61_0 * vx ||    by:=by + T_61_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
            )
            (branch " Case 2"
               (rule "modality_split_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3740741"))
               (rule "if_then_else_right" (formula "18") (quantifierEliminator "Mathematica") (nodenum "3740750"))
               (branch " by > Py"
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3740778"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3740786"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3740794"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3740802"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3740812"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3741100"))
                  (rule "ex_right" (formula "19") (newnames "T_62_0") (quantifierEliminator "Mathematica") (nodenum "3741304"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3741608"))
                  (branch " Case 1"
                     (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy >  0,  vy <  0,  by <= Py + paddleWidth,  T_62_0 >= 0   ")
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3741660"))
                     (branch " Case 1"
                        (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy >  0,  vy <  0,  by <= Py + paddleWidth,  \\forall R ts_88.     (   0 <= ts_88 & ts_88 <= T_62_0     ->       bx + ts_88 * vx >= 0            & bx + ts_88 * vx <= Width          & by + ts_88 * vy >= 0        & by + ts_88 * vy <= Height)   ")
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "4") (quantifierEliminator "Mathematica") (nodenum "3741700"))
                        (opengoal " by >  Py,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  vy >  0,  vy <  0,  by <= Py + paddleWidth,  {Oy:=by ||    Pvy:=Pvmax ||    Py:=Py + Pvmax * T_62_0 ||    bx:=Width + T_62_0 * vx ||    by:=by + T_62_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                     )
                  )
               )
               (branch " ! by > Py"
                  (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3741726"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3741735"))
                  (rule "modality_split_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3741743"))
                  (rule "assignment_to_update_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3741751"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3741759"))
                  (rule "ODESolve_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3741769"))
                  (builtin "Update Simplification" (formula "19") (nodenum "3742057"))
                  (rule "ex_right" (formula "19") (newnames "T_63_0") (quantifierEliminator "Mathematica") (nodenum "3742267"))
                  (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3745280"))
                  (branch " Case 1"
                     (builtin "Eliminate Universal Quantifiers" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3745333"))
                  )
                  (branch " Case 2"
                     (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3745335"))
                     (branch " Case 1"
                        (rule "all_right" (formula "19") (inst "skC=ts_94") (quantifierEliminator "Mathematica") (nodenum "3745370"))
                        (rule "imp_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3745511"))
                        (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3745548"))
                        (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3745596"))
                        (branch " Case 1"
                           (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3745642"))
                           (branch " Case 1"
                              (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3745690"))
                              (branch " Case 1"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3745753"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3745788"))
                              )
                              (branch " Case 2"
                                 (rule "applyEq" (formula "21") (term "0,0") (ifseqformula "5") (quantifierEliminator "Mathematica") (nodenum "3745800"))
                                 (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3745827"))
                              )
                           )
                           (branch " Case 2"
                              (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3745836"))
                           )
                        )
                        (branch " Case 2"
                           (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3745845"))
                        )
                     )
                     (branch " Case 2"
                        (rule "applyEq" (formula "19") (term "0,3") (ifseqformula "3") (quantifierEliminator "Mathematica") (nodenum "3745856"))
                        (rule "loop_unwind_dia" (formula "19") (term "5") (userinteraction) (quantifierEliminator "Mathematica") (nodenum "3746009"))
                        (builtin "Update Simplification" (formula "19") (nodenum "3746168"))
                        (rule "or_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3747356"))
                        (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3747390"))
                        (branch " Case 1"
                           (rule "and_right" (formula "19") (quantifierEliminator "Mathematica") (nodenum "3747432"))
                           (branch " Case 1"
                              (rule "modality_split_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3747495"))
                              (rule "if_then_else_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3747522"))
                              (branch " by = 0 & vy < 0"
                                 (builtin "Update Simplification" (formula "1") (nodenum "3747673"))
                                 (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3747713"))
                                 (rule "close_goal" (formula "19") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3747759"))
                              )
                              (branch " ! by = 0 & vy < 0"
                                 (builtin "Update Simplification" (formula "1") (nodenum "3747760"))
                                 (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3747780"))
                                 (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3747800"))
                                 (branch " Case 1"
                                    (rule "if_then_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3747832"))
                                    (branch " by = Height & vy > 0"
                                       (builtin "Update Simplification" (formula "1") (nodenum "3747985"))
                                       (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3748023"))
                                       (rule "close_goal" (formula "19") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3748072"))
                                    )
                                    (branch " ! by = Height & vy > 0"
                                       (builtin "Update Simplification" (formula "1") (nodenum "3748073"))
                                       (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3748093"))
                                       (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3748113"))
                                       (branch " Case 1"
                                          (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3748145"))
                                          (rule "if_then_else_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3748171"))
                                          (branch " by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3748274"))
                                             (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3748309"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3748335"))
                                             (rule "modality_split_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3748364"))
                                             (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3748391"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3748419"))
                                             (rule "ODESolve_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3748450"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3748738"))
                                             (rule "ex_right" (formula "23") (newnames "T_65_0") (quantifierEliminator "Mathematica") (nodenum "3749131"))
                                             (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3749524"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3749577"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3749579"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "23") (inst "skC=ts_93") (quantifierEliminator "Mathematica") (nodenum "3749614"))
                                                   (rule "imp_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3749759"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3749796"))
                                                   (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3749844"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3749890"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3749938"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3750008"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3750027"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3750044"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3750061"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy = Height,  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  Width + T_63_0 * vx >= 0,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_65_0 ||    bx:=Width + T_63_0 * vx + T_65_0 * vx ||    by:=by + T_63_0 * vy + T_65_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                          (branch " ! by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3750079"))
                                             (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3750107"))
                                             (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3750140"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3750172"))
                                             (rule "modality_split_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3750201"))
                                             (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3750224"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3750252"))
                                             (rule "ODESolve_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3750283"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3750571"))
                                             (rule "ex_right" (formula "23") (newnames "T_66_0") (quantifierEliminator "Mathematica") (nodenum "3750965"))
                                             (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3751360"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3751413"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3751415"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "23") (inst "skC=ts_96") (quantifierEliminator "Mathematica") (nodenum "3751450"))
                                                   (rule "imp_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3751595"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3751632"))
                                                   (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3751680"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3751726"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3751774"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3751844"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3751863"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3751880"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3751897"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by + T_63_0 * vy = Height,  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  Width + T_63_0 * vx >= 0,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_66_0 ||    bx:=Width + T_63_0 * vx + T_66_0 * vx ||    by:=by + T_63_0 * vy + T_66_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                       )
                                       (branch " Case 2"
                                          (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3751925"))
                                          (rule "if_then_else_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3751940"))
                                          (branch " by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3751967"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3751985"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3751993"))
                                             (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3752010"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3752025"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3752033"))
                                             (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3752052"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3752340"))
                                             (rule "ex_right" (formula "22") (newnames "T_67_0") (quantifierEliminator "Mathematica") (nodenum "3752733"))
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753126"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753179"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753181"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "22") (inst "skC=ts_98") (quantifierEliminator "Mathematica") (nodenum "3753216"))
                                                   (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753361"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3753398"))
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3753446"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3753492"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3753540"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3753610"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3753629"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3753646"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3753663"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  Width + T_63_0 * vx >= 0,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_67_0 ||    bx:=Width + T_63_0 * vx + T_67_0 * vx ||    by:=by + T_63_0 * vy + T_67_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                          (branch " ! by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3753681"))
                                             (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3753690"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753707"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3753715"))
                                             (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753734"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753749"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3753757"))
                                             (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3786063"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3786351"))
                                             (rule "ex_right" (formula "22") (newnames "T_86_0") (quantifierEliminator "Mathematica") (nodenum "3786745"))
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3787140"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3787193"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3787195"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "22") (inst "skC=ts_136") (quantifierEliminator "Mathematica") (nodenum "3787230"))
                                                   (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3787375"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3787412"))
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3787460"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3787506"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3787554"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3787624"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3787643"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3787660"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3787677"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  Width + T_63_0 * vx >= 0,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_86_0 ||    bx:=Width + T_63_0 * vx + T_86_0 * vx ||    by:=by + T_63_0 * vy + T_86_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                       )
                                    )
                                 )
                                 (branch " Case 2"
                                    (rule "if_then_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3753772"))
                                    (branch " by = Height & vy > 0"
                                       (builtin "Update Simplification" (formula "1") (nodenum "3753799"))
                                       (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3753808"))
                                       (rule "close_goal" (formula "18") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3753823"))
                                    )
                                    (branch " ! by = Height & vy > 0"
                                       (builtin "Update Simplification" (formula "1") (nodenum "3753824"))
                                       (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3753833"))
                                       (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3753843"))
                                       (branch " Case 1"
                                          (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3753860"))
                                          (rule "if_then_else_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3753877"))
                                          (branch " by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3753904"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753922"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3753930"))
                                             (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753947"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753962"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3753970"))
                                             (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3753989"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3754277"))
                                             (rule "ex_right" (formula "22") (newnames "T_68_0") (quantifierEliminator "Mathematica") (nodenum "3754670"))
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3755063"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3755116"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3755118"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "22") (inst "skC=ts_100") (quantifierEliminator "Mathematica") (nodenum "3755153"))
                                                   (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3755298"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3755335"))
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3755383"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3755429"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3755477"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3755547"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3755566"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3755583"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3755600"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy = Height,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  Width + T_63_0 * vx >= 0,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_68_0 ||    bx:=Width + T_63_0 * vx + T_68_0 * vx ||    by:=by + T_63_0 * vy + T_68_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                          (branch " ! by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3755618"))
                                             (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3755627"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3755644"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3755652"))
                                             (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3755671"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3755686"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3755694"))
                                             (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3755713"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3756001"))
                                             (rule "ex_right" (formula "22") (newnames "T_69_0") (quantifierEliminator "Mathematica") (nodenum "3756395"))
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3756790"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3756843"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3756845"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "22") (inst "skC=ts_102") (quantifierEliminator "Mathematica") (nodenum "3756880"))
                                                   (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3757025"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3757062"))
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3757110"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3757156"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3757204"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3757274"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3757293"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3757310"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3757327"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by + T_63_0 * vy = Height,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  Width + T_63_0 * vx >= 0,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_69_0 ||    bx:=Width + T_63_0 * vx + T_69_0 * vx ||    by:=by + T_63_0 * vy + T_69_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                       )
                                       (branch " Case 2"
                                          (rule "modality_split_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3757353"))
                                          (rule "if_then_else_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3757368"))
                                          (branch " by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3757395"))
                                             (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3757411"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3757419"))
                                             (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3757438"))
                                             (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3757453"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3757461"))
                                             (rule "ODESolve_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3757480"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3757768"))
                                             (rule "ex_right" (formula "21") (newnames "T_70_0") (quantifierEliminator "Mathematica") (nodenum "3758161"))
                                             (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3758554"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3758607"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3758609"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "21") (inst "skC=ts_104") (quantifierEliminator "Mathematica") (nodenum "3758644"))
                                                   (rule "imp_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3758789"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3758826"))
                                                   (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3758874"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3758920"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3758968"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3759038"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3759057"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3759074"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3759091"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  Width + T_63_0 * vx >= 0,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_70_0 ||    bx:=Width + T_63_0 * vx + T_70_0 * vx ||    by:=by + T_63_0 * vy + T_70_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                          (branch " ! by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3759109"))
                                             (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3759118"))
                                             (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3759135"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3759143"))
                                             (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3759162"))
                                             (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3759177"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3759185"))
                                             (rule "ODESolve_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3759204"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3759492"))
                                             (rule "ex_right" (formula "21") (newnames "T_71_0") (quantifierEliminator "Mathematica") (nodenum "3759886"))
                                             (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3760281"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3760334"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3760336"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "21") (inst "skC=ts_106") (quantifierEliminator "Mathematica") (nodenum "3760371"))
                                                   (rule "imp_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3760516"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3760553"))
                                                   (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3760601"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3760647"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3760695"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3787712"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3760766"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3760783"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3760800"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  Width + T_63_0 * vx >= 0,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_71_0 ||    bx:=Width + T_63_0 * vx + T_71_0 * vx ||    by:=by + T_63_0 * vy + T_71_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                       )
                                    )
                                 )
                              )
                           )
                           (branch " Case 2"
                              (rule "modality_split_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3760832"))
                              (rule "if_then_else_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3760851"))
                              (branch " by = 0 & vy < 0"
                                 (builtin "Update Simplification" (formula "1") (nodenum "3760878"))
                                 (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3760887"))
                                 (rule "close_goal" (formula "19") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3760902"))
                              )
                              (branch " ! by = 0 & vy < 0"
                                 (builtin "Update Simplification" (formula "1") (nodenum "3760903"))
                                 (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3760912"))
                                 (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3760922"))
                                 (branch " Case 1"
                                    (rule "if_then_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3760939"))
                                    (branch " by = Height & vy > 0"
                                       (builtin "Update Simplification" (formula "1") (nodenum "3760966"))
                                       (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3760975"))
                                       (rule "close_goal" (formula "19") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3760990"))
                                    )
                                    (branch " ! by = Height & vy > 0"
                                       (builtin "Update Simplification" (formula "1") (nodenum "3760991"))
                                       (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3761000"))
                                       (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3761010"))
                                       (branch " Case 1"
                                          (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3761029"))
                                          (rule "if_then_else_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3761046"))
                                          (branch " by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3761073"))
                                             (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3761093"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3761101"))
                                             (rule "modality_split_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3761116"))
                                             (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3761131"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3761139"))
                                             (rule "ODESolve_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3761158"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3761446"))
                                             (rule "ex_right" (formula "23") (newnames "T_72_0") (quantifierEliminator "Mathematica") (nodenum "3761839"))
                                             (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3762232"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3762285"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3762287"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "23") (inst "skC=ts_108") (quantifierEliminator "Mathematica") (nodenum "3762322"))
                                                   (rule "imp_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3762467"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3762504"))
                                                   (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3762552"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3762598"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3762646"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3762716"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3762735"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3762752"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3762769"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy = Height,  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy >= Py + -Pvmax * T_63_0 - paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_72_0 ||    bx:=Width + T_63_0 * vx + T_72_0 * vx ||    by:=by + T_63_0 * vy + T_72_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                          (branch " ! by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3762787"))
                                             (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3762796"))
                                             (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3762813"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3762821"))
                                             (rule "modality_split_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3762840"))
                                             (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3762855"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3762863"))
                                             (rule "ODESolve_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3762882"))
                                             (builtin "Update Simplification" (formula "23") (nodenum "3763170"))
                                             (rule "ex_right" (formula "23") (newnames "T_73_0") (quantifierEliminator "Mathematica") (nodenum "3763564"))
                                             (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3763959"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3764012"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3764014"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "23") (inst "skC=ts_110") (quantifierEliminator "Mathematica") (nodenum "3764049"))
                                                   (rule "imp_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3764194"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3764231"))
                                                   (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3764279"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3764325"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3764373"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3764443"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3764462"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3764479"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3764496"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by + T_63_0 * vy = Height,  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy >= Py + -Pvmax * T_63_0 - paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_73_0 ||    bx:=Width + T_63_0 * vx + T_73_0 * vx ||    by:=by + T_63_0 * vy + T_73_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                       )
                                       (branch " Case 2"
                                          (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3764524"))
                                          (rule "if_then_else_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3764539"))
                                          (branch " by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3764566"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3764584"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3764592"))
                                             (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3764609"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3764624"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3764632"))
                                             (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3764651"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3764939"))
                                             (rule "ex_right" (formula "22") (newnames "T_74_0") (quantifierEliminator "Mathematica") (nodenum "3765332"))
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3765725"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3765778"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3765780"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "22") (inst "skC=ts_112") (quantifierEliminator "Mathematica") (nodenum "3765815"))
                                                   (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3765960"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3765997"))
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3766045"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3766091"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3766139"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3766209"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3766228"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3766245"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3766262"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy >= Py + -Pvmax * T_63_0 - paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_74_0 ||    bx:=Width + T_63_0 * vx + T_74_0 * vx ||    by:=by + T_63_0 * vy + T_74_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                          (branch " ! by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3766280"))
                                             (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3766289"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3766306"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3766314"))
                                             (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3766333"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3766348"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3766356"))
                                             (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3787725"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3788013"))
                                             (rule "ex_right" (formula "22") (newnames "T_87_0") (quantifierEliminator "Mathematica") (nodenum "3788407"))
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3788802"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3788855"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3788857"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "22") (inst "skC=ts_138") (quantifierEliminator "Mathematica") (nodenum "3788892"))
                                                   (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3789037"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3789074"))
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3789122"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3789168"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3789216"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3789286"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3789305"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3789322"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3789339"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy >= Py + -Pvmax * T_63_0 - paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_87_0 ||    bx:=Width + T_63_0 * vx + T_87_0 * vx ||    by:=by + T_63_0 * vy + T_87_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                       )
                                    )
                                 )
                                 (branch " Case 2"
                                    (rule "if_then_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3766371"))
                                    (branch " by = Height & vy > 0"
                                       (builtin "Update Simplification" (formula "1") (nodenum "3766398"))
                                       (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3766407"))
                                       (rule "close_goal" (formula "18") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3766422"))
                                    )
                                    (branch " ! by = Height & vy > 0"
                                       (builtin "Update Simplification" (formula "1") (nodenum "3766423"))
                                       (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3766432"))
                                       (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3766442"))
                                       (branch " Case 1"
                                          (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3766459"))
                                          (rule "if_then_else_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3766476"))
                                          (branch " by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3766503"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3766521"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3766529"))
                                             (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3766546"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3766561"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3766569"))
                                             (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3766588"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3766876"))
                                             (rule "ex_right" (formula "22") (newnames "T_75_0") (quantifierEliminator "Mathematica") (nodenum "3767269"))
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3767662"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3767715"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3767717"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "22") (inst "skC=ts_114") (quantifierEliminator "Mathematica") (nodenum "3767752"))
                                                   (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3767897"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3767934"))
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3767982"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3768028"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3768076"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3768146"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3768165"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3768182"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3768199"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy = Height,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy >= Py + -Pvmax * T_63_0 - paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_75_0 ||    bx:=Width + T_63_0 * vx + T_75_0 * vx ||    by:=by + T_63_0 * vy + T_75_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                          (branch " ! by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3768217"))
                                             (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3768226"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3768243"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3768251"))
                                             (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3768270"))
                                             (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3768285"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3768293"))
                                             (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3768312"))
                                             (builtin "Update Simplification" (formula "22") (nodenum "3768600"))
                                             (rule "ex_right" (formula "22") (newnames "T_76_0") (quantifierEliminator "Mathematica") (nodenum "3768994"))
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3769389"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3769442"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3769444"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "22") (inst "skC=ts_116") (quantifierEliminator "Mathematica") (nodenum "3769479"))
                                                   (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3769624"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3769661"))
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3769709"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3769755"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3769803"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3769873"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3769892"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3769909"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3769926"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by + T_63_0 * vy = Height,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy >= Py + -Pvmax * T_63_0 - paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_76_0 ||    bx:=Width + T_63_0 * vx + T_76_0 * vx ||    by:=by + T_63_0 * vy + T_76_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                       )
                                       (branch " Case 2"
                                          (rule "modality_split_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3769952"))
                                          (rule "if_then_else_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3769967"))
                                          (branch " by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3769994"))
                                             (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3770014"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3770022"))
                                             (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3770037"))
                                             (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3770052"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3770060"))
                                             (rule "ODESolve_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3770079"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3770367"))
                                             (rule "ex_right" (formula "21") (newnames "T_77_0") (quantifierEliminator "Mathematica") (nodenum "3770760"))
                                             (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3771153"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3771206"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3771208"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "21") (inst "skC=ts_118") (quantifierEliminator "Mathematica") (nodenum "3771243"))
                                                   (rule "imp_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3771388"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3771425"))
                                                   (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3771473"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3771519"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3771567"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3771637"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3771656"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3771673"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3771690"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy >= Py + -Pvmax * T_63_0 - paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_77_0 ||    bx:=Width + T_63_0 * vx + T_77_0 * vx ||    by:=by + T_63_0 * vy + T_77_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                          (branch " ! by > Py"
                                             (builtin "Update Simplification" (formula "1") (nodenum "3771708"))
                                             (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3771717"))
                                             (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3771734"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3771742"))
                                             (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3771761"))
                                             (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3771776"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3771784"))
                                             (rule "ODESolve_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3771803"))
                                             (builtin "Update Simplification" (formula "21") (nodenum "3772091"))
                                             (rule "ex_right" (formula "21") (newnames "T_78_0") (quantifierEliminator "Mathematica") (nodenum "3772485"))
                                             (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3772880"))
                                             (branch " Case 1"
                                                (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3772933"))
                                             )
                                             (branch " Case 2"
                                                (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3772935"))
                                                (branch " Case 1"
                                                   (rule "all_right" (formula "21") (inst "skC=ts_120") (quantifierEliminator "Mathematica") (nodenum "3772970"))
                                                   (rule "imp_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3773115"))
                                                   (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3773152"))
                                                   (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3773200"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3773246"))
                                                      (branch " Case 1"
                                                         (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3773294"))
                                                         (branch " Case 1"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3789374"))
                                                         )
                                                         (branch " Case 2"
                                                            (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3773365"))
                                                         )
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3773382"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3773399"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy >= Py + -Pvmax * T_63_0 - paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_78_0 ||    bx:=Width + T_63_0 * vx + T_78_0 * vx ||    by:=by + T_63_0 * vy + T_78_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                                )
                                             )
                                          )
                                       )
                                    )
                                 )
                              )
                           )
                        )
                        (branch " Case 2"
                           (rule "modality_split_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3773433"))
                           (rule "if_then_else_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3773450"))
                           (branch " by = 0 & vy < 0"
                              (builtin "Update Simplification" (formula "1") (nodenum "3773477"))
                              (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3773486"))
                              (rule "close_goal" (formula "19") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3773501"))
                           )
                           (branch " ! by = 0 & vy < 0"
                              (builtin "Update Simplification" (formula "1") (nodenum "3773502"))
                              (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3773511"))
                              (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3773521"))
                              (branch " Case 1"
                                 (rule "if_then_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3773538"))
                                 (branch " by = Height & vy > 0"
                                    (builtin "Update Simplification" (formula "1") (nodenum "3773565"))
                                    (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3773574"))
                                    (rule "close_goal" (formula "19") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3773589"))
                                 )
                                 (branch " ! by = Height & vy > 0"
                                    (builtin "Update Simplification" (formula "1") (nodenum "3773590"))
                                    (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3773599"))
                                    (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3773609"))
                                    (branch " Case 1"
                                       (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3773628"))
                                       (rule "if_then_else_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3773645"))
                                       (branch " by > Py"
                                          (builtin "Update Simplification" (formula "1") (nodenum "3773672"))
                                          (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3773692"))
                                          (builtin "Update Simplification" (formula "23") (nodenum "3773700"))
                                          (rule "modality_split_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3773715"))
                                          (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3773730"))
                                          (builtin "Update Simplification" (formula "23") (nodenum "3773738"))
                                          (rule "ODESolve_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3773757"))
                                          (builtin "Update Simplification" (formula "23") (nodenum "3774045"))
                                          (rule "ex_right" (formula "23") (newnames "T_79_0") (quantifierEliminator "Mathematica") (nodenum "3774438"))
                                          (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3774831"))
                                          (branch " Case 1"
                                             (builtin "Eliminate Universal Quantifiers" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3774884"))
                                          )
                                          (branch " Case 2"
                                             (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3774886"))
                                             (branch " Case 1"
                                                (rule "all_right" (formula "23") (inst "skC=ts_122") (quantifierEliminator "Mathematica") (nodenum "3774921"))
                                                (rule "imp_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3775066"))
                                                (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3775103"))
                                                (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3775151"))
                                                (branch " Case 1"
                                                   (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3775197"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3775245"))
                                                      (branch " Case 1"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3775315"))
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3775334"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3775351"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3775368"))
                                                )
                                             )
                                             (branch " Case 2"
                                                (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy = Height,  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy <= Py + -Pvmax * T_63_0 + paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_79_0 ||    bx:=Width + T_63_0 * vx + T_79_0 * vx ||    by:=by + T_63_0 * vy + T_79_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                             )
                                          )
                                       )
                                       (branch " ! by > Py"
                                          (builtin "Update Simplification" (formula "1") (nodenum "3775386"))
                                          (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3775395"))
                                          (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3775412"))
                                          (builtin "Update Simplification" (formula "23") (nodenum "3775420"))
                                          (rule "modality_split_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3775439"))
                                          (rule "assignment_to_update_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3775454"))
                                          (builtin "Update Simplification" (formula "23") (nodenum "3775462"))
                                          (rule "ODESolve_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3775481"))
                                          (builtin "Update Simplification" (formula "23") (nodenum "3775769"))
                                          (rule "ex_right" (formula "23") (newnames "T_80_0") (quantifierEliminator "Mathematica") (nodenum "3776163"))
                                          (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3776558"))
                                          (branch " Case 1"
                                             (builtin "Eliminate Universal Quantifiers" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3776611"))
                                          )
                                          (branch " Case 2"
                                             (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3776613"))
                                             (branch " Case 1"
                                                (rule "all_right" (formula "23") (inst "skC=ts_124") (quantifierEliminator "Mathematica") (nodenum "3776648"))
                                                (rule "imp_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3776793"))
                                                (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3776830"))
                                                (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3776878"))
                                                (branch " Case 1"
                                                   (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3776924"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "25") (quantifierEliminator "Mathematica") (nodenum "3776972"))
                                                      (branch " Case 1"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3777042"))
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3777061"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3777078"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3777095"))
                                                )
                                             )
                                             (branch " Case 2"
                                                (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by + T_63_0 * vy = Height,  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy <= Py + -Pvmax * T_63_0 + paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_80_0 ||    bx:=Width + T_63_0 * vx + T_80_0 * vx ||    by:=by + T_63_0 * vy + T_80_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                             )
                                          )
                                       )
                                    )
                                    (branch " Case 2"
                                       (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3777123"))
                                       (rule "if_then_else_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3777138"))
                                       (branch " by > Py"
                                          (builtin "Update Simplification" (formula "1") (nodenum "3777165"))
                                          (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3777183"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3777191"))
                                          (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3777208"))
                                          (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3777223"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3777231"))
                                          (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3777250"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3777538"))
                                          (rule "ex_right" (formula "22") (newnames "T_81_0") (quantifierEliminator "Mathematica") (nodenum "3777931"))
                                          (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3778324"))
                                          (branch " Case 1"
                                             (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3778377"))
                                          )
                                          (branch " Case 2"
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3778379"))
                                             (branch " Case 1"
                                                (rule "all_right" (formula "22") (inst "skC=ts_126") (quantifierEliminator "Mathematica") (nodenum "3778414"))
                                                (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3778559"))
                                                (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3778596"))
                                                (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3778644"))
                                                (branch " Case 1"
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3778690"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3778738"))
                                                      (branch " Case 1"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3778808"))
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3778827"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3778844"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3778861"))
                                                )
                                             )
                                             (branch " Case 2"
                                                (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy <= Py + -Pvmax * T_63_0 + paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_81_0 ||    bx:=Width + T_63_0 * vx + T_81_0 * vx ||    by:=by + T_63_0 * vy + T_81_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                             )
                                          )
                                       )
                                       (branch " ! by > Py"
                                          (builtin "Update Simplification" (formula "1") (nodenum "3778879"))
                                          (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3778888"))
                                          (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3778905"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3778913"))
                                          (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3778932"))
                                          (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3778947"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3778955"))
                                          (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3789387"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3789675"))
                                          (rule "ex_right" (formula "22") (newnames "T_88_0") (quantifierEliminator "Mathematica") (nodenum "3790069"))
                                          (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3790464"))
                                          (branch " Case 1"
                                             (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3790517"))
                                          )
                                          (branch " Case 2"
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3790519"))
                                             (branch " Case 1"
                                                (rule "all_right" (formula "22") (inst "skC=ts_140") (quantifierEliminator "Mathematica") (nodenum "3790554"))
                                                (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3790699"))
                                                (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3790736"))
                                                (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3790819"))
                                                (branch " Case 1"
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3790865"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3790913"))
                                                      (branch " Case 1"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3790983"))
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3791002"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3791019"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3791036"))
                                                )
                                             )
                                             (branch " Case 2"
                                                (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by + T_63_0 * vy = 0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy <= Py + -Pvmax * T_63_0 + paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_88_0 ||    bx:=Width + T_63_0 * vx + T_88_0 * vx ||    by:=by + T_63_0 * vy + T_88_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                             )
                                          )
                                       )
                                    )
                                 )
                              )
                              (branch " Case 2"
                                 (rule "if_then_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3778970"))
                                 (branch " by = Height & vy > 0"
                                    (builtin "Update Simplification" (formula "1") (nodenum "3778997"))
                                    (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3779006"))
                                    (rule "close_goal" (formula "18") (ifseqformula "2") (quantifierEliminator "Mathematica") (nodenum "3779021"))
                                 )
                                 (branch " ! by = Height & vy > 0"
                                    (builtin "Update Simplification" (formula "1") (nodenum "3779022"))
                                    (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3779031"))
                                    (rule "and_right" (formula "15") (quantifierEliminator "Mathematica") (nodenum "3779041"))
                                    (branch " Case 1"
                                       (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3779058"))
                                       (rule "if_then_else_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3779075"))
                                       (branch " by > Py"
                                          (builtin "Update Simplification" (formula "1") (nodenum "3779102"))
                                          (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3779120"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3779128"))
                                          (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3779145"))
                                          (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3779160"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3779168"))
                                          (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3779187"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3779475"))
                                          (rule "ex_right" (formula "22") (newnames "T_82_0") (quantifierEliminator "Mathematica") (nodenum "3779868"))
                                          (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3780261"))
                                          (branch " Case 1"
                                             (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3780314"))
                                          )
                                          (branch " Case 2"
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3780316"))
                                             (branch " Case 1"
                                                (rule "all_right" (formula "22") (inst "skC=ts_128") (quantifierEliminator "Mathematica") (nodenum "3780351"))
                                                (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3780496"))
                                                (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3780533"))
                                                (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3780581"))
                                                (branch " Case 1"
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3780627"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3780675"))
                                                      (branch " Case 1"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3780745"))
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3780764"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3780781"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3780798"))
                                                )
                                             )
                                             (branch " Case 2"
                                                (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy = Height,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy <= Py + -Pvmax * T_63_0 + paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_82_0 ||    bx:=Width + T_63_0 * vx + T_82_0 * vx ||    by:=by + T_63_0 * vy + T_82_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                             )
                                          )
                                       )
                                       (branch " ! by > Py"
                                          (builtin "Update Simplification" (formula "1") (nodenum "3780816"))
                                          (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3780825"))
                                          (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3780842"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3780850"))
                                          (rule "modality_split_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3780869"))
                                          (rule "assignment_to_update_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3780884"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3780892"))
                                          (rule "ODESolve_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3780911"))
                                          (builtin "Update Simplification" (formula "22") (nodenum "3781199"))
                                          (rule "ex_right" (formula "22") (newnames "T_83_0") (quantifierEliminator "Mathematica") (nodenum "3781593"))
                                          (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3781988"))
                                          (branch " Case 1"
                                             (builtin "Eliminate Universal Quantifiers" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3782041"))
                                          )
                                          (branch " Case 2"
                                             (rule "and_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3782043"))
                                             (branch " Case 1"
                                                (rule "all_right" (formula "22") (inst "skC=ts_130") (quantifierEliminator "Mathematica") (nodenum "3782078"))
                                                (rule "imp_right" (formula "22") (quantifierEliminator "Mathematica") (nodenum "3782223"))
                                                (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3782260"))
                                                (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3782308"))
                                                (branch " Case 1"
                                                   (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3782354"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "24") (quantifierEliminator "Mathematica") (nodenum "3782402"))
                                                      (branch " Case 1"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3782472"))
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3782491"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3782508"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3782525"))
                                                )
                                             )
                                             (branch " Case 2"
                                                (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by + T_63_0 * vy = Height,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy <= Py + -Pvmax * T_63_0 + paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_83_0 ||    bx:=Width + T_63_0 * vx + T_83_0 * vx ||    by:=by + T_63_0 * vy + T_83_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                             )
                                          )
                                       )
                                    )
                                    (branch " Case 2"
                                       (rule "modality_split_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3782551"))
                                       (rule "if_then_else_right" (formula "20") (quantifierEliminator "Mathematica") (nodenum "3782566"))
                                       (branch " by > Py"
                                          (builtin "Update Simplification" (formula "1") (nodenum "3782593"))
                                          (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3782613"))
                                          (builtin "Update Simplification" (formula "21") (nodenum "3782621"))
                                          (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3782636"))
                                          (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3782651"))
                                          (builtin "Update Simplification" (formula "21") (nodenum "3782659"))
                                          (rule "ODESolve_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3782678"))
                                          (builtin "Update Simplification" (formula "21") (nodenum "3782966"))
                                          (rule "ex_right" (formula "21") (newnames "T_84_0") (quantifierEliminator "Mathematica") (nodenum "3783359"))
                                          (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3783752"))
                                          (branch " Case 1"
                                             (builtin "Eliminate Universal Quantifiers" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3783805"))
                                          )
                                          (branch " Case 2"
                                             (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3783807"))
                                             (branch " Case 1"
                                                (rule "all_right" (formula "21") (inst "skC=ts_132") (quantifierEliminator "Mathematica") (nodenum "3783842"))
                                                (rule "imp_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3783987"))
                                                (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3784024"))
                                                (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3784072"))
                                                (branch " Case 1"
                                                   (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3784118"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3784166"))
                                                      (branch " Case 1"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3784236"))
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3784255"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3784272"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3784289"))
                                                )
                                             )
                                             (branch " Case 2"
                                                (opengoal " by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy <= Py + -Pvmax * T_63_0 + paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=Pvmax ||    Py:=Py + -Pvmax * T_63_0 + Pvmax * T_84_0 ||    bx:=Width + T_63_0 * vx + T_84_0 * vx ||    by:=by + T_63_0 * vy + T_84_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                             )
                                          )
                                       )
                                       (branch " ! by > Py"
                                          (builtin "Update Simplification" (formula "1") (nodenum "3784307"))
                                          (rule "not_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3784316"))
                                          (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3784333"))
                                          (builtin "Update Simplification" (formula "21") (nodenum "3784341"))
                                          (rule "modality_split_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3784360"))
                                          (rule "assignment_to_update_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3784375"))
                                          (builtin "Update Simplification" (formula "21") (nodenum "3784383"))
                                          (rule "ODESolve_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3784402"))
                                          (builtin "Update Simplification" (formula "21") (nodenum "3784690"))
                                          (rule "ex_right" (formula "21") (newnames "T_85_0") (quantifierEliminator "Mathematica") (nodenum "3785084"))
                                          (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3785479"))
                                          (branch " Case 1"
                                             (builtin "Eliminate Universal Quantifiers" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3785532"))
                                          )
                                          (branch " Case 2"
                                             (rule "and_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3785534"))
                                             (branch " Case 1"
                                                (rule "all_right" (formula "21") (inst "skC=ts_134") (quantifierEliminator "Mathematica") (nodenum "3785569"))
                                                (rule "imp_right" (formula "21") (quantifierEliminator "Mathematica") (nodenum "3785714"))
                                                (rule "and_left" (formula "1") (quantifierEliminator "Mathematica") (nodenum "3785751"))
                                                (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3785799"))
                                                (branch " Case 1"
                                                   (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3785845"))
                                                   (branch " Case 1"
                                                      (rule "and_right" (formula "23") (quantifierEliminator "Mathematica") (nodenum "3785893"))
                                                      (branch " Case 1"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3790817"))
                                                      )
                                                      (branch " Case 2"
                                                         (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3785964"))
                                                      )
                                                   )
                                                   (branch " Case 2"
                                                      (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3785981"))
                                                   )
                                                )
                                                (branch " Case 2"
                                                   (builtin "Eliminate Universal Quantifiers" (formula "3") (quantifierEliminator "Mathematica") (nodenum "3785998"))
                                                )
                                             )
                                             (branch " Case 2"
                                                (opengoal " Width >  0,  Height >  0,  bx = Width,  by >  0,  vx <  0,  by <  Height,  vx ^ 2 + vy ^ 2 = (vel) ^ 2,  vel >  0,  paddleWidth >  0,  Pvmax ^ 2 >  vy ^ 2,  Pvmax >  0,  -Height * (Width / vel) <= Pvmax,  Py <  Height,  Py >  0 ==>  by + T_63_0 * vy >  Py + -Pvmax * T_63_0,  by >  Py,  vy >  0,  vy <  0,  by <= Py + paddleWidth,  by + T_63_0 * vy <= Py + -Pvmax * T_63_0 + paddleWidth,  {Oy:=by + T_63_0 * vy ||    Pvy:=-Pvmax ||    Py:=Py + -Pvmax * T_63_0 + -Pvmax * T_85_0 ||    bx:=Width + T_63_0 * vx + T_85_0 * vx ||    by:=by + T_63_0 * vy + T_85_0 * vy}    \\<      (if(by = 0 & vy < 0)        then          vy := -vy        else          if(by = Height & vy > 0)          then            vy := -vy          fi        fi ;        if(by > Py)        then          Pvy := Pvmax        else          Pvy := -Pvmax        fi ;        Oy := by ;        {bx' = vx, by' = vy,         Py' = Pvy, bx >= 0,         bx <= Width, by >= 0,         by <= Height})*      \\> (  bx >= 0 & by >= Py - paddleWidth          & by <= Py + paddleWidth)   ")
                                             )
                                          )
                                       )
                                    )
                                 )
                              )
                           )
                        )
                     )
                  )
               )
            )
         )
      )
   )
)
)

}
