DV0019

divisor_mask_prefix_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Ordinary prefix induction constructs every finite divisor mask inside the actual source domain, with explicit beta extensions at each step.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic statement

forall N F n l. (exists dst_positive_code_exists_input dst_positive_scale_exists_input dst_negative_code_exists_input dst_negative_scale_exists_input. (((F) = (((((dst_positive_code_exists_input) + (dst_positive_scale_exists_input)) * S ((dst_positive_code_exists_input) + (dst_positive_scale_exists_input)) + ((dst_positive_scale_exists_input) + (dst_positive_scale_exists_input))) + (((dst_negative_code_exists_input) + (dst_negative_scale_exists_input)) * S ((dst_negative_code_exists_input) + (dst_negative_scale_exists_input)) + ((dst_negative_scale_exists_input) + (dst_negative_scale_exists_input)))) * S ((((dst_positive_code_exists_input) + (dst_positive_scale_exists_input)) * S ((dst_positive_code_exists_input) + (dst_positive_scale_exists_input)) + ((dst_positive_scale_exists_input) + (dst_positive_scale_exists_input))) + (((dst_negative_code_exists_input) + (dst_negative_scale_exists_input)) * S ((dst_negative_code_exists_input) + (dst_negative_scale_exists_input)) + ((dst_negative_scale_exists_input) + (dst_negative_scale_exists_input)))) + ((((dst_negative_code_exists_input) + (dst_negative_scale_exists_input)) * S ((dst_negative_code_exists_input) + (dst_negative_scale_exists_input)) + ((dst_negative_scale_exists_input) + (dst_negative_scale_exists_input))) + (((dst_negative_code_exists_input) + (dst_negative_scale_exists_input)) * S ((dst_negative_code_exists_input) + (dst_negative_scale_exists_input)) + ((dst_negative_scale_exists_input) + (dst_negative_scale_exists_input)))))) /\ (forall dst_index_exists_input. (exists pvs_le_gap_exists_inputdomain. pvs_le_gap_exists_inputdomain + (dst_index_exists_input) = (N)) -> exists dst_positive_exists_input dst_negative_exists_input dst_value_exists_input. ((((exists ff_h_pvs_exists_inputentrypositive. ff_h_pvs_exists_inputentrypositive + S (dst_positive_exists_input) = S ((S (dst_index_exists_input)) * dst_positive_scale_exists_input)) /\ exists ff_q_pvs_exists_inputentrypositive. dst_positive_code_exists_input = ff_q_pvs_exists_inputentrypositive * S ((S (dst_index_exists_input)) * dst_positive_scale_exists_input) + (dst_positive_exists_input))) /\ (((((exists ff_h_pvs_exists_inputentrynegative. ff_h_pvs_exists_inputentrynegative + S (dst_negative_exists_input) = S ((S (dst_index_exists_input)) * dst_negative_scale_exists_input)) /\ exists ff_q_pvs_exists_inputentrynegative. dst_negative_code_exists_input = ff_q_pvs_exists_inputentrynegative * S ((S (dst_index_exists_input)) * dst_negative_scale_exists_input) + (dst_negative_exists_input))) /\ (exists ge_balance_positive_exists_inputentryvalue ge_balance_negative_exists_inputentryvalue. (((((dst_value_exists_input) = 2 * (ge_balance_positive_exists_inputentryvalue) /\ (ge_balance_negative_exists_inputentryvalue) = 0) \/ exists ge_signed_half_exists_inputentryvaluedecode. (((dst_value_exists_input) = 2 * ge_signed_half_exists_inputentryvaluedecode + 1 /\ (ge_balance_positive_exists_inputentryvalue) = 0) /\ (ge_balance_negative_exists_inputentryvalue) = S ge_signed_half_exists_inputentryvaluedecode))) /\ ((dst_positive_exists_input) + ge_balance_negative_exists_inputentryvalue = (dst_negative_exists_input) + ge_balance_positive_exists_inputentryvalue))))))))) -> (exists pvs_le_gap_exists_bound. pvs_le_gap_exists_bound + (l) = (N)) -> exists M. (((exists dst_positive_code_exists_masktable dst_positive_scale_exists_masktable dst_negative_code_exists_masktable dst_negative_scale_exists_masktable. (((M) = (((((dst_positive_code_exists_masktable) + (dst_positive_scale_exists_masktable)) * S ((dst_positive_code_exists_masktable) + (dst_positive_scale_exists_masktable)) + ((dst_positive_scale_exists_masktable) + (dst_positive_scale_exists_masktable))) + (((dst_negative_code_exists_masktable) + (dst_negative_scale_exists_masktable)) * S ((dst_negative_code_exists_masktable) + (dst_negative_scale_exists_masktable)) + ((dst_negative_scale_exists_masktable) + (dst_negative_scale_exists_masktable)))) * S ((((dst_positive_code_exists_masktable) + (dst_positive_scale_exists_masktable)) * S ((dst_positive_code_exists_masktable) + (dst_positive_scale_exists_masktable)) + ((dst_positive_scale_exists_masktable) + (dst_positive_scale_exists_masktable))) + (((dst_negative_code_exists_masktable) + (dst_negative_scale_exists_masktable)) * S ((dst_negative_code_exists_masktable) + (dst_negative_scale_exists_masktable)) + ((dst_negative_scale_exists_masktable) + (dst_negative_scale_exists_masktable)))) + ((((dst_negative_code_exists_masktable) + (dst_negative_scale_exists_masktable)) * S ((dst_negative_code_exists_masktable) + (dst_negative_scale_exists_masktable)) + ((dst_negative_scale_exists_masktable) + (dst_negative_scale_exists_masktable))) + (((dst_negative_code_exists_masktable) + (dst_negative_scale_exists_masktable)) * S ((dst_negative_code_exists_masktable) + (dst_negative_scale_exists_masktable)) + ((dst_negative_scale_exists_masktable) + (dst_negative_scale_exists_masktable)))))) /\ (forall dst_index_exists_masktable. (exists pvs_le_gap_exists_masktabledomain. pvs_le_gap_exists_masktabledomain + (dst_index_exists_masktable) = (l)) -> exists dst_positive_exists_masktable dst_negative_exists_masktable dst_value_exists_masktable. ((((exists ff_h_pvs_exists_masktableentrypositive. ff_h_pvs_exists_masktableentrypositive + S (dst_positive_exists_masktable) = S ((S (dst_index_exists_masktable)) * dst_positive_scale_exists_masktable)) /\ exists ff_q_pvs_exists_masktableentrypositive. dst_positive_code_exists_masktable = ff_q_pvs_exists_masktableentrypositive * S ((S (dst_index_exists_masktable)) * dst_positive_scale_exists_masktable) + (dst_positive_exists_masktable))) /\ (((((exists ff_h_pvs_exists_masktableentrynegative. ff_h_pvs_exists_masktableentrynegative + S (dst_negative_exists_masktable) = S ((S (dst_index_exists_masktable)) * dst_negative_scale_exists_masktable)) /\ exists ff_q_pvs_exists_masktableentrynegative. dst_negative_code_exists_masktable = ff_q_pvs_exists_masktableentrynegative * S ((S (dst_index_exists_masktable)) * dst_negative_scale_exists_masktable) + (dst_negative_exists_masktable))) /\ (exists ge_balance_positive_exists_masktableentryvalue ge_balance_negative_exists_masktableentryvalue. (((((dst_value_exists_masktable) = 2 * (ge_balance_positive_exists_masktableentryvalue) /\ (ge_balance_negative_exists_masktableentryvalue) = 0) \/ exists ge_signed_half_exists_masktableentryvaluedecode. (((dst_value_exists_masktable) = 2 * ge_signed_half_exists_masktableentryvaluedecode + 1 /\ (ge_balance_positive_exists_masktableentryvalue) = 0) /\ (ge_balance_negative_exists_masktableentryvalue) = S ge_signed_half_exists_masktableentryvaluedecode))) /\ ((dst_positive_exists_masktable) + ge_balance_negative_exists_masktableentryvalue = (dst_negative_exists_masktable) + ge_balance_positive_exists_masktableentryvalue))))))))) /\ (forall dm_index_exists_mask dm_value_exists_mask. (exists pvs_le_gap_exists_maskdomain. pvs_le_gap_exists_maskdomain + (dm_index_exists_mask) = (l)) -> (exists dst_positive_code_exists_masklookup dst_positive_scale_exists_masklookup dst_negative_code_exists_masklookup dst_negative_scale_exists_masklookup dst_positive_exists_masklookup dst_negative_exists_masklookup. (((M) = (((((dst_positive_code_exists_masklookup) + (dst_positive_scale_exists_masklookup)) * S ((dst_positive_code_exists_masklookup) + (dst_positive_scale_exists_masklookup)) + ((dst_positive_scale_exists_masklookup) + (dst_positive_scale_exists_masklookup))) + (((dst_negative_code_exists_masklookup) + (dst_negative_scale_exists_masklookup)) * S ((dst_negative_code_exists_masklookup) + (dst_negative_scale_exists_masklookup)) + ((dst_negative_scale_exists_masklookup) + (dst_negative_scale_exists_masklookup)))) * S ((((dst_positive_code_exists_masklookup) + (dst_positive_scale_exists_masklookup)) * S ((dst_positive_code_exists_masklookup) + (dst_positive_scale_exists_masklookup)) + ((dst_positive_scale_exists_masklookup) + (dst_positive_scale_exists_masklookup))) + (((dst_negative_code_exists_masklookup) + (dst_negative_scale_exists_masklookup)) * S ((dst_negative_code_exists_masklookup) + (dst_negative_scale_exists_masklookup)) + ((dst_negative_scale_exists_masklookup) + (dst_negative_scale_exists_masklookup)))) + ((((dst_negative_code_exists_masklookup) + (dst_negative_scale_exists_masklookup)) * S ((dst_negative_code_exists_masklookup) + (dst_negative_scale_exists_masklookup)) + ((dst_negative_scale_exists_masklookup) + (dst_negative_scale_exists_masklookup))) + (((dst_negative_code_exists_masklookup) + (dst_negative_scale_exists_masklookup)) * S ((dst_negative_code_exists_masklookup) + (dst_negative_scale_exists_masklookup)) + ((dst_negative_scale_exists_masklookup) + (dst_negative_scale_exists_masklookup)))))) /\ (((((exists ff_h_pvs_exists_masklookuppositive. ff_h_pvs_exists_masklookuppositive + S (dst_positive_exists_masklookup) = S ((S (dm_index_exists_mask)) * dst_positive_scale_exists_masklookup)) /\ exists ff_q_pvs_exists_masklookuppositive. dst_positive_code_exists_masklookup = ff_q_pvs_exists_masklookuppositive * S ((S (dm_index_exists_mask)) * dst_positive_scale_exists_masklookup) + (dst_positive_exists_masklookup))) /\ (((((exists ff_h_pvs_exists_masklookupnegative. ff_h_pvs_exists_masklookupnegative + S (dst_negative_exists_masklookup) = S ((S (dm_index_exists_mask)) * dst_negative_scale_exists_masklookup)) /\ exists ff_q_pvs_exists_masklookupnegative. dst_negative_code_exists_masklookup = ff_q_pvs_exists_masklookupnegative * S ((S (dm_index_exists_mask)) * dst_negative_scale_exists_masklookup) + (dst_negative_exists_masklookup))) /\ (exists ge_balance_positive_exists_masklookupvalue ge_balance_negative_exists_masklookupvalue. (((((dm_value_exists_mask) = 2 * (ge_balance_positive_exists_masklookupvalue) /\ (ge_balance_negative_exists_masklookupvalue) = 0) \/ exists ge_signed_half_exists_masklookupvaluedecode. (((dm_value_exists_mask) = 2 * ge_signed_half_exists_masklookupvaluedecode + 1 /\ (ge_balance_positive_exists_masklookupvalue) = 0) /\ (ge_balance_negative_exists_masklookupvalue) = S ge_signed_half_exists_masklookupvaluedecode))) /\ ((dst_positive_exists_masklookup) + ge_balance_negative_exists_masklookupvalue = (dst_negative_exists_masklookup) + ge_balance_positive_exists_masklookupvalue))))))))) -> ((((~((dm_index_exists_mask)=0)) /\ (exists dm_quotient_exists_maskentry. (((n)=(dm_index_exists_mask)*dm_quotient_exists_maskentry) /\ (exists dst_positive_code_exists_maskentryinput dst_positive_scale_exists_maskentryinput dst_negative_code_exists_maskentryinput dst_negative_scale_exists_maskentryinput dst_positive_exists_maskentryinput dst_negative_exists_maskentryinput. (((F) = (((((dst_positive_code_exists_maskentryinput) + (dst_positive_scale_exists_maskentryinput)) * S ((dst_positive_code_exists_maskentryinput) + (dst_positive_scale_exists_maskentryinput)) + ((dst_positive_scale_exists_maskentryinput) + (dst_positive_scale_exists_maskentryinput))) + (((dst_negative_code_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)) * S ((dst_negative_code_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)) + ((dst_negative_scale_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)))) * S ((((dst_positive_code_exists_maskentryinput) + (dst_positive_scale_exists_maskentryinput)) * S ((dst_positive_code_exists_maskentryinput) + (dst_positive_scale_exists_maskentryinput)) + ((dst_positive_scale_exists_maskentryinput) + (dst_positive_scale_exists_maskentryinput))) + (((dst_negative_code_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)) * S ((dst_negative_code_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)) + ((dst_negative_scale_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)))) + ((((dst_negative_code_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)) * S ((dst_negative_code_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)) + ((dst_negative_scale_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput))) + (((dst_negative_code_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)) * S ((dst_negative_code_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)) + ((dst_negative_scale_exists_maskentryinput) + (dst_negative_scale_exists_maskentryinput)))))) /\ (((((exists ff_h_pvs_exists_maskentryinputpositive. ff_h_pvs_exists_maskentryinputpositive + S (dst_positive_exists_maskentryinput) = S ((S (dm_index_exists_mask)) * dst_positive_scale_exists_maskentryinput)) /\ exists ff_q_pvs_exists_maskentryinputpositive. dst_positive_code_exists_maskentryinput = ff_q_pvs_exists_maskentryinputpositive * S ((S (dm_index_exists_mask)) * dst_positive_scale_exists_maskentryinput) + (dst_positive_exists_maskentryinput))) /\ (((((exists ff_h_pvs_exists_maskentryinputnegative. ff_h_pvs_exists_maskentryinputnegative + S (dst_negative_exists_maskentryinput) = S ((S (dm_index_exists_mask)) * dst_negative_scale_exists_maskentryinput)) /\ exists ff_q_pvs_exists_maskentryinputnegative. dst_negative_code_exists_maskentryinput = ff_q_pvs_exists_maskentryinputnegative * S ((S (dm_index_exists_mask)) * dst_negative_scale_exists_maskentryinput) + (dst_negative_exists_maskentryinput))) /\ (exists ge_balance_positive_exists_maskentryinputvalue ge_balance_negative_exists_maskentryinputvalue. (((((dm_value_exists_mask) = 2 * (ge_balance_positive_exists_maskentryinputvalue) /\ (ge_balance_negative_exists_maskentryinputvalue) = 0) \/ exists ge_signed_half_exists_maskentryinputvaluedecode. (((dm_value_exists_mask) = 2 * ge_signed_half_exists_maskentryinputvaluedecode + 1 /\ (ge_balance_positive_exists_maskentryinputvalue) = 0) /\ (ge_balance_negative_exists_maskentryinputvalue) = S ge_signed_half_exists_maskentryinputvaluedecode))) /\ ((dst_positive_exists_maskentryinput) + ge_balance_negative_exists_maskentryinputvalue = (dst_negative_exists_maskentryinput) + ge_balance_positive_exists_maskentryinputvalue))))))))))))) \/ ((((dm_index_exists_mask)=0 \/ ~(exists pvs_factor_exists_maskentrynondivisor. (n) = (dm_index_exists_mask) * pvs_factor_exists_maskentrynondivisor)) /\ ((dm_value_exists_mask)=0)))))))

Constructive proof overview

Generated structural guide

Ordinary prefix induction constructs every finite divisor mask inside the actual source domain, with explicit beta extensions at each step.

The unchanged tactic script uses 6 declared prerequisites and contains 56 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

DV0005 arithmetic_signed_table_singleton DV0017 divisor_mask_prefix_zero_constructor le_trans Stable theorem; checked-use authorized le_succ_self Stable theorem; checked-use authorized DV0013 divisor_mask_entry_exists DV0018 divisor_mask_prefix_append

Direct dependents

DV0020 signed_divisor_sum_exists DV0024 signed_divisor_sum_one

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

56 script commands · 16 reading checkpoints · 5 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Named ingredients (4)
arithmetic signed table singletondivisor mask prefix zero constructordivisor mask entry existsdivisor mask prefix append

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–4

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro N
  2. L2
    intro F
  3. L3
    intro n
  4. L4
    intro l
02Induction on lL5–7

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L5
    induction l
  2. L6
    intro ht
  3. L7
    intro hbound
03Establish hzeroL8–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply arithmetic signed table singleton.

  1. L8
    have hzero : ∃ M. ArithTable(0,M) ∧ ArithAt(M,0,0)Definitions: ArithTableArithAt
  2. L9
    specialize arithmetic_signed_table_singleton (0)
  3. L10
    apply arithmetic_signed_table_singleton
04Separate the logical casesL11–12

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L11
    cases hzero
  2. L12
    cases hzero_witness
05Construct an explicit witnessL13–13

Supply the displayed value, then prove that it has the required property.

  1. L13
    exists x
06Use earlier factsL14–19

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L14
    specialize divisor_mask_prefix_zero_constructor (F)
  2. L15
    specialize divisor_mask_prefix_zero_constructor (n)
  3. L16
    specialize divisor_mask_prefix_zero_constructor (x)
  4. L17
    apply divisor_mask_prefix_zero_constructor
  5. L18
    exact hzero_witness_left
  6. L19
    exact hzero_witness_right
07Fix variables and assumptionsL20–21

Work with arbitrary variables or the premises of the current implication.

  1. L20
    intro ht
  2. L21
    intro hbound
08Establish hlowL22–29

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le trans.

  1. L22
    have hlow : exists pvs_le_gap_exists_previous_bound. pvs_le_gap_exists_previous_bound + (l) = (N)
  2. L23
    specialize le_trans (l)
  3. L24
    specialize le_trans (S l)
  4. L25
    specialize le_trans (N)
  5. L26
    apply le_trans
  6. L27
    specialize le_succ_self (l)
  7. L28
    apply le_succ_self
  8. L29
    exact hbound
09Establish hprevL30–33

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply IH.

  1. L30
    have hprev : ∃ M. DivisorMask(F,n,l,M)Definitions: DivisorMask
  2. L31
    apply IH
  3. L32
    exact ht
  4. L33
    exact hlow
10Separate the logical casesL34–34

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L34
    cases hprev
11Establish hzL35–42

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply divisor mask entry exists.

  1. L35
    have hz : ∃ z. DivisorMaskEntry(F,n,S l,z)Definitions: DivisorMaskEntry
  2. L36
    specialize divisor_mask_entry_exists (N)
  3. L37
    specialize divisor_mask_entry_exists (F)
  4. L38
    specialize divisor_mask_entry_exists (n)
  5. L39
    specialize divisor_mask_entry_exists (S l)
  6. L40
    apply divisor_mask_entry_exists
  7. L41
    exact ht
  8. L42
    exact hbound
12Separate the logical casesL43–43

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L43
    cases hz
13Establish hnextL44–52

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply divisor mask prefix append.

  1. L44
    have hnext : ∃ G. DivisorMask(F,n,S l,G) ∧ ArithTableEqual(x,G,S l)Definitions: ArithTableEqualDivisorMask
  2. L45
    specialize divisor_mask_prefix_append (F)
  3. L46
    specialize divisor_mask_prefix_append (n)
  4. L47
    specialize divisor_mask_prefix_append (l)
  5. L48
    specialize divisor_mask_prefix_append (x)
  6. L49
    specialize divisor_mask_prefix_append (x1)
  7. L50
    apply divisor_mask_prefix_append
  8. L51
    exact hprev_witness
  9. L52
    exact hz_witness
14Separate the logical casesL53–54

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L53
    cases hnext
  2. L54
    cases hnext_witness
15Construct an explicit witnessL55–55

Supply the displayed value, then prove that it has the required property.

  1. L55
    exists x2
16Use earlier factsL56–56

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L56
    exact hnext_witness_left

Library-wide reading audit

Original exact command ledger · 56 lines
  1. 0001intro N
  2. 0002intro F
  3. 0003intro n
  4. 0004intro l
  5. 0005induction l
  6. 0006intro ht
  7. 0007intro hbound
  8. 0008have hzero : exists M. (exists dst_positive_code_exists_base_table dst_positive_scale_exists_base_table dst_negative_code_exists_base_table dst_negative_scale_exists_base_table. (((M) = (((((dst_positive_code_exists_base_table) + (dst_positive_scale_exists_base_table)) * S ((dst_positive_code_exists_base_table) + (dst_positive_scale_exists_base_table)) + ((dst_positive_scale_exists_base_table) + (dst_positive_scale_exists_base_table))) + (((dst_negative_code_exists_base_table) + (dst_negative_scale_exists_base_table)) * S ((dst_negative_code_exists_base_table) + (dst_negative_scale_exists_base_table)) + ((dst_negative_scale_exists_base_table) + (dst_negative_scale_exists_base_table)))) * S ((((dst_positive_code_exists_base_table) + (dst_positive_scale_exists_base_table)) * S ((dst_positive_code_exists_base_table) + (dst_positive_scale_exists_base_table)) + ((dst_positive_scale_exists_base_table) + (dst_positive_scale_exists_base_table))) + (((dst_negative_code_exists_base_table) + (dst_negative_scale_exists_base_table)) * S ((dst_negative_code_exists_base_table) + (dst_negative_scale_exists_base_table)) + ((dst_negative_scale_exists_base_table) + (dst_negative_scale_exists_base_table)))) + ((((dst_negative_code_exists_base_table) + (dst_negative_scale_exists_base_table)) * S ((dst_negative_code_exists_base_table) + (dst_negative_scale_exists_base_table)) + ((dst_negative_scale_exists_base_table) + (dst_negative_scale_exists_base_table))) + (((dst_negative_code_exists_base_table) + (dst_negative_scale_exists_base_table)) * S ((dst_negative_code_exists_base_table) + (dst_negative_scale_exists_base_table)) + ((dst_negative_scale_exists_base_table) + (dst_negative_scale_exists_base_table)))))) /\ (forall dst_index_exists_base_table. (exists pvs_le_gap_exists_base_tabledomain. pvs_le_gap_exists_base_tabledomain + (dst_index_exists_base_table) = (0)) -> exists dst_positive_exists_base_table dst_negative_exists_base_table dst_value_exists_base_table. ((((exists ff_h_pvs_exists_base_tableentrypositive. ff_h_pvs_exists_base_tableentrypositive + S (dst_positive_exists_base_table) = S ((S (dst_index_exists_base_table)) * dst_positive_scale_exists_base_table)) /\ exists ff_q_pvs_exists_base_tableentrypositive. dst_positive_code_exists_base_table = ff_q_pvs_exists_base_tableentrypositive * S ((S (dst_index_exists_base_table)) * dst_positive_scale_exists_base_table) + (dst_positive_exists_base_table))) /\ (((((exists ff_h_pvs_exists_base_tableentrynegative. ff_h_pvs_exists_base_tableentrynegative + S (dst_negative_exists_base_table) = S ((S (dst_index_exists_base_table)) * dst_negative_scale_exists_base_table)) /\ exists ff_q_pvs_exists_base_tableentrynegative. dst_negative_code_exists_base_table = ff_q_pvs_exists_base_tableentrynegative * S ((S (dst_index_exists_base_table)) * dst_negative_scale_exists_base_table) + (dst_negative_exists_base_table))) /\ (exists ge_balance_positive_exists_base_tableentryvalue ge_balance_negative_exists_base_tableentryvalue. (((((dst_value_exists_base_table) = 2 * (ge_balance_positive_exists_base_tableentryvalue) /\ (ge_balance_negative_exists_base_tableentryvalue) = 0) \/ exists ge_signed_half_exists_base_tableentryvaluedecode. (((dst_value_exists_base_table) = 2 * ge_signed_half_exists_base_tableentryvaluedecode + 1 /\ (ge_balance_positive_exists_base_tableentryvalue) = 0) /\ (ge_balance_negative_exists_base_tableentryvalue) = S ge_signed_half_exists_base_tableentryvaluedecode))) /\ ((dst_positive_exists_base_table) + ge_balance_negative_exists_base_tableentryvalue = (dst_negative_exists_base_table) + ge_balance_positive_exists_base_tableentryvalue))))))))) /\ (exists dst_positive_code_exists_base_entry dst_positive_scale_exists_base_entry dst_negative_code_exists_base_entry dst_negative_scale_exists_base_entry dst_positive_exists_base_entry dst_negative_exists_base_entry. (((M) = (((((dst_positive_code_exists_base_entry) + (dst_positive_scale_exists_base_entry)) * S ((dst_positive_code_exists_base_entry) + (dst_positive_scale_exists_base_entry)) + ((dst_positive_scale_exists_base_entry) + (dst_positive_scale_exists_base_entry))) + (((dst_negative_code_exists_base_entry) + (dst_negative_scale_exists_base_entry)) * S ((dst_negative_code_exists_base_entry) + (dst_negative_scale_exists_base_entry)) + ((dst_negative_scale_exists_base_entry) + (dst_negative_scale_exists_base_entry)))) * S ((((dst_positive_code_exists_base_entry) + (dst_positive_scale_exists_base_entry)) * S ((dst_positive_code_exists_base_entry) + (dst_positive_scale_exists_base_entry)) + ((dst_positive_scale_exists_base_entry) + (dst_positive_scale_exists_base_entry))) + (((dst_negative_code_exists_base_entry) + (dst_negative_scale_exists_base_entry)) * S ((dst_negative_code_exists_base_entry) + (dst_negative_scale_exists_base_entry)) + ((dst_negative_scale_exists_base_entry) + (dst_negative_scale_exists_base_entry)))) + ((((dst_negative_code_exists_base_entry) + (dst_negative_scale_exists_base_entry)) * S ((dst_negative_code_exists_base_entry) + (dst_negative_scale_exists_base_entry)) + ((dst_negative_scale_exists_base_entry) + (dst_negative_scale_exists_base_entry))) + (((dst_negative_code_exists_base_entry) + (dst_negative_scale_exists_base_entry)) * S ((dst_negative_code_exists_base_entry) + (dst_negative_scale_exists_base_entry)) + ((dst_negative_scale_exists_base_entry) + (dst_negative_scale_exists_base_entry)))))) /\ (((((exists ff_h_pvs_exists_base_entrypositive. ff_h_pvs_exists_base_entrypositive + S (dst_positive_exists_base_entry) = S ((S (0)) * dst_positive_scale_exists_base_entry)) /\ exists ff_q_pvs_exists_base_entrypositive. dst_positive_code_exists_base_entry = ff_q_pvs_exists_base_entrypositive * S ((S (0)) * dst_positive_scale_exists_base_entry) + (dst_positive_exists_base_entry))) /\ (((((exists ff_h_pvs_exists_base_entrynegative. ff_h_pvs_exists_base_entrynegative + S (dst_negative_exists_base_entry) = S ((S (0)) * dst_negative_scale_exists_base_entry)) /\ exists ff_q_pvs_exists_base_entrynegative. dst_negative_code_exists_base_entry = ff_q_pvs_exists_base_entrynegative * S ((S (0)) * dst_negative_scale_exists_base_entry) + (dst_negative_exists_base_entry))) /\ (exists ge_balance_positive_exists_base_entryvalue ge_balance_negative_exists_base_entryvalue. (((((0) = 2 * (ge_balance_positive_exists_base_entryvalue) /\ (ge_balance_negative_exists_base_entryvalue) = 0) \/ exists ge_signed_half_exists_base_entryvaluedecode. (((0) = 2 * ge_signed_half_exists_base_entryvaluedecode + 1 /\ (ge_balance_positive_exists_base_entryvalue) = 0) /\ (ge_balance_negative_exists_base_entryvalue) = S ge_signed_half_exists_base_entryvaluedecode))) /\ ((dst_positive_exists_base_entry) + ge_balance_negative_exists_base_entryvalue = (dst_negative_exists_base_entry) + ge_balance_positive_exists_base_entryvalue)))))))))
  9. 0009specialize arithmetic_signed_table_singleton (0)
  10. 0010apply arithmetic_signed_table_singleton
  11. 0011cases hzero
  12. 0012cases hzero_witness
  13. 0013exists x
  14. 0014specialize divisor_mask_prefix_zero_constructor (F)
  15. 0015specialize divisor_mask_prefix_zero_constructor (n)
  16. 0016specialize divisor_mask_prefix_zero_constructor (x)
  17. 0017apply divisor_mask_prefix_zero_constructor
  18. 0018exact hzero_witness_left
  19. 0019exact hzero_witness_right
  20. 0020intro ht
  21. 0021intro hbound
  22. 0022have hlow : exists pvs_le_gap_exists_previous_bound. pvs_le_gap_exists_previous_bound + (l) = (N)
  23. 0023specialize le_trans (l)
  24. 0024specialize le_trans (S l)
  25. 0025specialize le_trans (N)
  26. 0026apply le_trans
  27. 0027specialize le_succ_self (l)
  28. 0028apply le_succ_self
  29. 0029exact hbound
  30. 0030have hprev : exists M. (((exists dst_positive_code_exists_previous_masktable dst_positive_scale_exists_previous_masktable dst_negative_code_exists_previous_masktable dst_negative_scale_exists_previous_masktable. (((M) = (((((dst_positive_code_exists_previous_masktable) + (dst_positive_scale_exists_previous_masktable)) * S ((dst_positive_code_exists_previous_masktable) + (dst_positive_scale_exists_previous_masktable)) + ((dst_positive_scale_exists_previous_masktable) + (dst_positive_scale_exists_previous_masktable))) + (((dst_negative_code_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)) * S ((dst_negative_code_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)) + ((dst_negative_scale_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)))) * S ((((dst_positive_code_exists_previous_masktable) + (dst_positive_scale_exists_previous_masktable)) * S ((dst_positive_code_exists_previous_masktable) + (dst_positive_scale_exists_previous_masktable)) + ((dst_positive_scale_exists_previous_masktable) + (dst_positive_scale_exists_previous_masktable))) + (((dst_negative_code_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)) * S ((dst_negative_code_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)) + ((dst_negative_scale_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)))) + ((((dst_negative_code_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)) * S ((dst_negative_code_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)) + ((dst_negative_scale_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable))) + (((dst_negative_code_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)) * S ((dst_negative_code_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)) + ((dst_negative_scale_exists_previous_masktable) + (dst_negative_scale_exists_previous_masktable)))))) /\ (forall dst_index_exists_previous_masktable. (exists pvs_le_gap_exists_previous_masktabledomain. pvs_le_gap_exists_previous_masktabledomain + (dst_index_exists_previous_masktable) = (l)) -> exists dst_positive_exists_previous_masktable dst_negative_exists_previous_masktable dst_value_exists_previous_masktable. ((((exists ff_h_pvs_exists_previous_masktableentrypositive. ff_h_pvs_exists_previous_masktableentrypositive + S (dst_positive_exists_previous_masktable) = S ((S (dst_index_exists_previous_masktable)) * dst_positive_scale_exists_previous_masktable)) /\ exists ff_q_pvs_exists_previous_masktableentrypositive. dst_positive_code_exists_previous_masktable = ff_q_pvs_exists_previous_masktableentrypositive * S ((S (dst_index_exists_previous_masktable)) * dst_positive_scale_exists_previous_masktable) + (dst_positive_exists_previous_masktable))) /\ (((((exists ff_h_pvs_exists_previous_masktableentrynegative. ff_h_pvs_exists_previous_masktableentrynegative + S (dst_negative_exists_previous_masktable) = S ((S (dst_index_exists_previous_masktable)) * dst_negative_scale_exists_previous_masktable)) /\ exists ff_q_pvs_exists_previous_masktableentrynegative. dst_negative_code_exists_previous_masktable = ff_q_pvs_exists_previous_masktableentrynegative * S ((S (dst_index_exists_previous_masktable)) * dst_negative_scale_exists_previous_masktable) + (dst_negative_exists_previous_masktable))) /\ (exists ge_balance_positive_exists_previous_masktableentryvalue ge_balance_negative_exists_previous_masktableentryvalue. (((((dst_value_exists_previous_masktable) = 2 * (ge_balance_positive_exists_previous_masktableentryvalue) /\ (ge_balance_negative_exists_previous_masktableentryvalue) = 0) \/ exists ge_signed_half_exists_previous_masktableentryvaluedecode. (((dst_value_exists_previous_masktable) = 2 * ge_signed_half_exists_previous_masktableentryvaluedecode + 1 /\ (ge_balance_positive_exists_previous_masktableentryvalue) = 0) /\ (ge_balance_negative_exists_previous_masktableentryvalue) = S ge_signed_half_exists_previous_masktableentryvaluedecode))) /\ ((dst_positive_exists_previous_masktable) + ge_balance_negative_exists_previous_masktableentryvalue = (dst_negative_exists_previous_masktable) + ge_balance_positive_exists_previous_masktableentryvalue))))))))) /\ (forall dm_index_exists_previous_mask dm_value_exists_previous_mask. (exists pvs_le_gap_exists_previous_maskdomain. pvs_le_gap_exists_previous_maskdomain + (dm_index_exists_previous_mask) = (l)) -> (exists dst_positive_code_exists_previous_masklookup dst_positive_scale_exists_previous_masklookup dst_negative_code_exists_previous_masklookup dst_negative_scale_exists_previous_masklookup dst_positive_exists_previous_masklookup dst_negative_exists_previous_masklookup. (((M) = (((((dst_positive_code_exists_previous_masklookup) + (dst_positive_scale_exists_previous_masklookup)) * S ((dst_positive_code_exists_previous_masklookup) + (dst_positive_scale_exists_previous_masklookup)) + ((dst_positive_scale_exists_previous_masklookup) + (dst_positive_scale_exists_previous_masklookup))) + (((dst_negative_code_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)) * S ((dst_negative_code_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)) + ((dst_negative_scale_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)))) * S ((((dst_positive_code_exists_previous_masklookup) + (dst_positive_scale_exists_previous_masklookup)) * S ((dst_positive_code_exists_previous_masklookup) + (dst_positive_scale_exists_previous_masklookup)) + ((dst_positive_scale_exists_previous_masklookup) + (dst_positive_scale_exists_previous_masklookup))) + (((dst_negative_code_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)) * S ((dst_negative_code_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)) + ((dst_negative_scale_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)))) + ((((dst_negative_code_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)) * S ((dst_negative_code_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)) + ((dst_negative_scale_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup))) + (((dst_negative_code_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)) * S ((dst_negative_code_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)) + ((dst_negative_scale_exists_previous_masklookup) + (dst_negative_scale_exists_previous_masklookup)))))) /\ (((((exists ff_h_pvs_exists_previous_masklookuppositive. ff_h_pvs_exists_previous_masklookuppositive + S (dst_positive_exists_previous_masklookup) = S ((S (dm_index_exists_previous_mask)) * dst_positive_scale_exists_previous_masklookup)) /\ exists ff_q_pvs_exists_previous_masklookuppositive. dst_positive_code_exists_previous_masklookup = ff_q_pvs_exists_previous_masklookuppositive * S ((S (dm_index_exists_previous_mask)) * dst_positive_scale_exists_previous_masklookup) + (dst_positive_exists_previous_masklookup))) /\ (((((exists ff_h_pvs_exists_previous_masklookupnegative. ff_h_pvs_exists_previous_masklookupnegative + S (dst_negative_exists_previous_masklookup) = S ((S (dm_index_exists_previous_mask)) * dst_negative_scale_exists_previous_masklookup)) /\ exists ff_q_pvs_exists_previous_masklookupnegative. dst_negative_code_exists_previous_masklookup = ff_q_pvs_exists_previous_masklookupnegative * S ((S (dm_index_exists_previous_mask)) * dst_negative_scale_exists_previous_masklookup) + (dst_negative_exists_previous_masklookup))) /\ (exists ge_balance_positive_exists_previous_masklookupvalue ge_balance_negative_exists_previous_masklookupvalue. (((((dm_value_exists_previous_mask) = 2 * (ge_balance_positive_exists_previous_masklookupvalue) /\ (ge_balance_negative_exists_previous_masklookupvalue) = 0) \/ exists ge_signed_half_exists_previous_masklookupvaluedecode. (((dm_value_exists_previous_mask) = 2 * ge_signed_half_exists_previous_masklookupvaluedecode + 1 /\ (ge_balance_positive_exists_previous_masklookupvalue) = 0) /\ (ge_balance_negative_exists_previous_masklookupvalue) = S ge_signed_half_exists_previous_masklookupvaluedecode))) /\ ((dst_positive_exists_previous_masklookup) + ge_balance_negative_exists_previous_masklookupvalue = (dst_negative_exists_previous_masklookup) + ge_balance_positive_exists_previous_masklookupvalue))))))))) -> ((((~((dm_index_exists_previous_mask)=0)) /\ (exists dm_quotient_exists_previous_maskentry. (((n)=(dm_index_exists_previous_mask)*dm_quotient_exists_previous_maskentry) /\ (exists dst_positive_code_exists_previous_maskentryinput dst_positive_scale_exists_previous_maskentryinput dst_negative_code_exists_previous_maskentryinput dst_negative_scale_exists_previous_maskentryinput dst_positive_exists_previous_maskentryinput dst_negative_exists_previous_maskentryinput. (((F) = (((((dst_positive_code_exists_previous_maskentryinput) + (dst_positive_scale_exists_previous_maskentryinput)) * S ((dst_positive_code_exists_previous_maskentryinput) + (dst_positive_scale_exists_previous_maskentryinput)) + ((dst_positive_scale_exists_previous_maskentryinput) + (dst_positive_scale_exists_previous_maskentryinput))) + (((dst_negative_code_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)) * S ((dst_negative_code_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)) + ((dst_negative_scale_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)))) * S ((((dst_positive_code_exists_previous_maskentryinput) + (dst_positive_scale_exists_previous_maskentryinput)) * S ((dst_positive_code_exists_previous_maskentryinput) + (dst_positive_scale_exists_previous_maskentryinput)) + ((dst_positive_scale_exists_previous_maskentryinput) + (dst_positive_scale_exists_previous_maskentryinput))) + (((dst_negative_code_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)) * S ((dst_negative_code_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)) + ((dst_negative_scale_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)))) + ((((dst_negative_code_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)) * S ((dst_negative_code_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)) + ((dst_negative_scale_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput))) + (((dst_negative_code_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)) * S ((dst_negative_code_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)) + ((dst_negative_scale_exists_previous_maskentryinput) + (dst_negative_scale_exists_previous_maskentryinput)))))) /\ (((((exists ff_h_pvs_exists_previous_maskentryinputpositive. ff_h_pvs_exists_previous_maskentryinputpositive + S (dst_positive_exists_previous_maskentryinput) = S ((S (dm_index_exists_previous_mask)) * dst_positive_scale_exists_previous_maskentryinput)) /\ exists ff_q_pvs_exists_previous_maskentryinputpositive. dst_positive_code_exists_previous_maskentryinput = ff_q_pvs_exists_previous_maskentryinputpositive * S ((S (dm_index_exists_previous_mask)) * dst_positive_scale_exists_previous_maskentryinput) + (dst_positive_exists_previous_maskentryinput))) /\ (((((exists ff_h_pvs_exists_previous_maskentryinputnegative. ff_h_pvs_exists_previous_maskentryinputnegative + S (dst_negative_exists_previous_maskentryinput) = S ((S (dm_index_exists_previous_mask)) * dst_negative_scale_exists_previous_maskentryinput)) /\ exists ff_q_pvs_exists_previous_maskentryinputnegative. dst_negative_code_exists_previous_maskentryinput = ff_q_pvs_exists_previous_maskentryinputnegative * S ((S (dm_index_exists_previous_mask)) * dst_negative_scale_exists_previous_maskentryinput) + (dst_negative_exists_previous_maskentryinput))) /\ (exists ge_balance_positive_exists_previous_maskentryinputvalue ge_balance_negative_exists_previous_maskentryinputvalue. (((((dm_value_exists_previous_mask) = 2 * (ge_balance_positive_exists_previous_maskentryinputvalue) /\ (ge_balance_negative_exists_previous_maskentryinputvalue) = 0) \/ exists ge_signed_half_exists_previous_maskentryinputvaluedecode. (((dm_value_exists_previous_mask) = 2 * ge_signed_half_exists_previous_maskentryinputvaluedecode + 1 /\ (ge_balance_positive_exists_previous_maskentryinputvalue) = 0) /\ (ge_balance_negative_exists_previous_maskentryinputvalue) = S ge_signed_half_exists_previous_maskentryinputvaluedecode))) /\ ((dst_positive_exists_previous_maskentryinput) + ge_balance_negative_exists_previous_maskentryinputvalue = (dst_negative_exists_previous_maskentryinput) + ge_balance_positive_exists_previous_maskentryinputvalue))))))))))))) \/ ((((dm_index_exists_previous_mask)=0 \/ ~(exists pvs_factor_exists_previous_maskentrynondivisor. (n) = (dm_index_exists_previous_mask) * pvs_factor_exists_previous_maskentrynondivisor)) /\ ((dm_value_exists_previous_mask)=0)))))))
  31. 0031apply IH
  32. 0032exact ht
  33. 0033exact hlow
  34. 0034cases hprev
  35. 0035have hz : exists z. ((((~((S l)=0)) /\ (exists dm_quotient_exists_next_value. (((n)=(S l)*dm_quotient_exists_next_value) /\ (exists dst_positive_code_exists_next_valueinput dst_positive_scale_exists_next_valueinput dst_negative_code_exists_next_valueinput dst_negative_scale_exists_next_valueinput dst_positive_exists_next_valueinput dst_negative_exists_next_valueinput. (((F) = (((((dst_positive_code_exists_next_valueinput) + (dst_positive_scale_exists_next_valueinput)) * S ((dst_positive_code_exists_next_valueinput) + (dst_positive_scale_exists_next_valueinput)) + ((dst_positive_scale_exists_next_valueinput) + (dst_positive_scale_exists_next_valueinput))) + (((dst_negative_code_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)) * S ((dst_negative_code_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)) + ((dst_negative_scale_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)))) * S ((((dst_positive_code_exists_next_valueinput) + (dst_positive_scale_exists_next_valueinput)) * S ((dst_positive_code_exists_next_valueinput) + (dst_positive_scale_exists_next_valueinput)) + ((dst_positive_scale_exists_next_valueinput) + (dst_positive_scale_exists_next_valueinput))) + (((dst_negative_code_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)) * S ((dst_negative_code_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)) + ((dst_negative_scale_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)))) + ((((dst_negative_code_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)) * S ((dst_negative_code_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)) + ((dst_negative_scale_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput))) + (((dst_negative_code_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)) * S ((dst_negative_code_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)) + ((dst_negative_scale_exists_next_valueinput) + (dst_negative_scale_exists_next_valueinput)))))) /\ (((((exists ff_h_pvs_exists_next_valueinputpositive. ff_h_pvs_exists_next_valueinputpositive + S (dst_positive_exists_next_valueinput) = S ((S (S l)) * dst_positive_scale_exists_next_valueinput)) /\ exists ff_q_pvs_exists_next_valueinputpositive. dst_positive_code_exists_next_valueinput = ff_q_pvs_exists_next_valueinputpositive * S ((S (S l)) * dst_positive_scale_exists_next_valueinput) + (dst_positive_exists_next_valueinput))) /\ (((((exists ff_h_pvs_exists_next_valueinputnegative. ff_h_pvs_exists_next_valueinputnegative + S (dst_negative_exists_next_valueinput) = S ((S (S l)) * dst_negative_scale_exists_next_valueinput)) /\ exists ff_q_pvs_exists_next_valueinputnegative. dst_negative_code_exists_next_valueinput = ff_q_pvs_exists_next_valueinputnegative * S ((S (S l)) * dst_negative_scale_exists_next_valueinput) + (dst_negative_exists_next_valueinput))) /\ (exists ge_balance_positive_exists_next_valueinputvalue ge_balance_negative_exists_next_valueinputvalue. (((((z) = 2 * (ge_balance_positive_exists_next_valueinputvalue) /\ (ge_balance_negative_exists_next_valueinputvalue) = 0) \/ exists ge_signed_half_exists_next_valueinputvaluedecode. (((z) = 2 * ge_signed_half_exists_next_valueinputvaluedecode + 1 /\ (ge_balance_positive_exists_next_valueinputvalue) = 0) /\ (ge_balance_negative_exists_next_valueinputvalue) = S ge_signed_half_exists_next_valueinputvaluedecode))) /\ ((dst_positive_exists_next_valueinput) + ge_balance_negative_exists_next_valueinputvalue = (dst_negative_exists_next_valueinput) + ge_balance_positive_exists_next_valueinputvalue))))))))))))) \/ ((((S l)=0 \/ ~(exists pvs_factor_exists_next_valuenondivisor. (n) = (S l) * pvs_factor_exists_next_valuenondivisor)) /\ ((z)=0))))
  36. 0036specialize divisor_mask_entry_exists (N)
  37. 0037specialize divisor_mask_entry_exists (F)
  38. 0038specialize divisor_mask_entry_exists (n)
  39. 0039specialize divisor_mask_entry_exists (S l)
  40. 0040apply divisor_mask_entry_exists
  41. 0041exact ht
  42. 0042exact hbound
  43. 0043cases hz
  44. 0044have hnext : exists G. (((exists dst_positive_code_exists_next_masktable dst_positive_scale_exists_next_masktable dst_negative_code_exists_next_masktable dst_negative_scale_exists_next_masktable. (((G) = (((((dst_positive_code_exists_next_masktable) + (dst_positive_scale_exists_next_masktable)) * S ((dst_positive_code_exists_next_masktable) + (dst_positive_scale_exists_next_masktable)) + ((dst_positive_scale_exists_next_masktable) + (dst_positive_scale_exists_next_masktable))) + (((dst_negative_code_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)) * S ((dst_negative_code_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)) + ((dst_negative_scale_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)))) * S ((((dst_positive_code_exists_next_masktable) + (dst_positive_scale_exists_next_masktable)) * S ((dst_positive_code_exists_next_masktable) + (dst_positive_scale_exists_next_masktable)) + ((dst_positive_scale_exists_next_masktable) + (dst_positive_scale_exists_next_masktable))) + (((dst_negative_code_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)) * S ((dst_negative_code_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)) + ((dst_negative_scale_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)))) + ((((dst_negative_code_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)) * S ((dst_negative_code_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)) + ((dst_negative_scale_exists_next_masktable) + (dst_negative_scale_exists_next_masktable))) + (((dst_negative_code_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)) * S ((dst_negative_code_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)) + ((dst_negative_scale_exists_next_masktable) + (dst_negative_scale_exists_next_masktable)))))) /\ (forall dst_index_exists_next_masktable. (exists pvs_le_gap_exists_next_masktabledomain. pvs_le_gap_exists_next_masktabledomain + (dst_index_exists_next_masktable) = (S l)) -> exists dst_positive_exists_next_masktable dst_negative_exists_next_masktable dst_value_exists_next_masktable. ((((exists ff_h_pvs_exists_next_masktableentrypositive. ff_h_pvs_exists_next_masktableentrypositive + S (dst_positive_exists_next_masktable) = S ((S (dst_index_exists_next_masktable)) * dst_positive_scale_exists_next_masktable)) /\ exists ff_q_pvs_exists_next_masktableentrypositive. dst_positive_code_exists_next_masktable = ff_q_pvs_exists_next_masktableentrypositive * S ((S (dst_index_exists_next_masktable)) * dst_positive_scale_exists_next_masktable) + (dst_positive_exists_next_masktable))) /\ (((((exists ff_h_pvs_exists_next_masktableentrynegative. ff_h_pvs_exists_next_masktableentrynegative + S (dst_negative_exists_next_masktable) = S ((S (dst_index_exists_next_masktable)) * dst_negative_scale_exists_next_masktable)) /\ exists ff_q_pvs_exists_next_masktableentrynegative. dst_negative_code_exists_next_masktable = ff_q_pvs_exists_next_masktableentrynegative * S ((S (dst_index_exists_next_masktable)) * dst_negative_scale_exists_next_masktable) + (dst_negative_exists_next_masktable))) /\ (exists ge_balance_positive_exists_next_masktableentryvalue ge_balance_negative_exists_next_masktableentryvalue. (((((dst_value_exists_next_masktable) = 2 * (ge_balance_positive_exists_next_masktableentryvalue) /\ (ge_balance_negative_exists_next_masktableentryvalue) = 0) \/ exists ge_signed_half_exists_next_masktableentryvaluedecode. (((dst_value_exists_next_masktable) = 2 * ge_signed_half_exists_next_masktableentryvaluedecode + 1 /\ (ge_balance_positive_exists_next_masktableentryvalue) = 0) /\ (ge_balance_negative_exists_next_masktableentryvalue) = S ge_signed_half_exists_next_masktableentryvaluedecode))) /\ ((dst_positive_exists_next_masktable) + ge_balance_negative_exists_next_masktableentryvalue = (dst_negative_exists_next_masktable) + ge_balance_positive_exists_next_masktableentryvalue))))))))) /\ (forall dm_index_exists_next_mask dm_value_exists_next_mask. (exists pvs_le_gap_exists_next_maskdomain. pvs_le_gap_exists_next_maskdomain + (dm_index_exists_next_mask) = (S l)) -> (exists dst_positive_code_exists_next_masklookup dst_positive_scale_exists_next_masklookup dst_negative_code_exists_next_masklookup dst_negative_scale_exists_next_masklookup dst_positive_exists_next_masklookup dst_negative_exists_next_masklookup. (((G) = (((((dst_positive_code_exists_next_masklookup) + (dst_positive_scale_exists_next_masklookup)) * S ((dst_positive_code_exists_next_masklookup) + (dst_positive_scale_exists_next_masklookup)) + ((dst_positive_scale_exists_next_masklookup) + (dst_positive_scale_exists_next_masklookup))) + (((dst_negative_code_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)) * S ((dst_negative_code_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)) + ((dst_negative_scale_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)))) * S ((((dst_positive_code_exists_next_masklookup) + (dst_positive_scale_exists_next_masklookup)) * S ((dst_positive_code_exists_next_masklookup) + (dst_positive_scale_exists_next_masklookup)) + ((dst_positive_scale_exists_next_masklookup) + (dst_positive_scale_exists_next_masklookup))) + (((dst_negative_code_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)) * S ((dst_negative_code_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)) + ((dst_negative_scale_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)))) + ((((dst_negative_code_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)) * S ((dst_negative_code_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)) + ((dst_negative_scale_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup))) + (((dst_negative_code_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)) * S ((dst_negative_code_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)) + ((dst_negative_scale_exists_next_masklookup) + (dst_negative_scale_exists_next_masklookup)))))) /\ (((((exists ff_h_pvs_exists_next_masklookuppositive. ff_h_pvs_exists_next_masklookuppositive + S (dst_positive_exists_next_masklookup) = S ((S (dm_index_exists_next_mask)) * dst_positive_scale_exists_next_masklookup)) /\ exists ff_q_pvs_exists_next_masklookuppositive. dst_positive_code_exists_next_masklookup = ff_q_pvs_exists_next_masklookuppositive * S ((S (dm_index_exists_next_mask)) * dst_positive_scale_exists_next_masklookup) + (dst_positive_exists_next_masklookup))) /\ (((((exists ff_h_pvs_exists_next_masklookupnegative. ff_h_pvs_exists_next_masklookupnegative + S (dst_negative_exists_next_masklookup) = S ((S (dm_index_exists_next_mask)) * dst_negative_scale_exists_next_masklookup)) /\ exists ff_q_pvs_exists_next_masklookupnegative. dst_negative_code_exists_next_masklookup = ff_q_pvs_exists_next_masklookupnegative * S ((S (dm_index_exists_next_mask)) * dst_negative_scale_exists_next_masklookup) + (dst_negative_exists_next_masklookup))) /\ (exists ge_balance_positive_exists_next_masklookupvalue ge_balance_negative_exists_next_masklookupvalue. (((((dm_value_exists_next_mask) = 2 * (ge_balance_positive_exists_next_masklookupvalue) /\ (ge_balance_negative_exists_next_masklookupvalue) = 0) \/ exists ge_signed_half_exists_next_masklookupvaluedecode. (((dm_value_exists_next_mask) = 2 * ge_signed_half_exists_next_masklookupvaluedecode + 1 /\ (ge_balance_positive_exists_next_masklookupvalue) = 0) /\ (ge_balance_negative_exists_next_masklookupvalue) = S ge_signed_half_exists_next_masklookupvaluedecode))) /\ ((dst_positive_exists_next_masklookup) + ge_balance_negative_exists_next_masklookupvalue = (dst_negative_exists_next_masklookup) + ge_balance_positive_exists_next_masklookupvalue))))))))) -> ((((~((dm_index_exists_next_mask)=0)) /\ (exists dm_quotient_exists_next_maskentry. (((n)=(dm_index_exists_next_mask)*dm_quotient_exists_next_maskentry) /\ (exists dst_positive_code_exists_next_maskentryinput dst_positive_scale_exists_next_maskentryinput dst_negative_code_exists_next_maskentryinput dst_negative_scale_exists_next_maskentryinput dst_positive_exists_next_maskentryinput dst_negative_exists_next_maskentryinput. (((F) = (((((dst_positive_code_exists_next_maskentryinput) + (dst_positive_scale_exists_next_maskentryinput)) * S ((dst_positive_code_exists_next_maskentryinput) + (dst_positive_scale_exists_next_maskentryinput)) + ((dst_positive_scale_exists_next_maskentryinput) + (dst_positive_scale_exists_next_maskentryinput))) + (((dst_negative_code_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)) * S ((dst_negative_code_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)) + ((dst_negative_scale_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)))) * S ((((dst_positive_code_exists_next_maskentryinput) + (dst_positive_scale_exists_next_maskentryinput)) * S ((dst_positive_code_exists_next_maskentryinput) + (dst_positive_scale_exists_next_maskentryinput)) + ((dst_positive_scale_exists_next_maskentryinput) + (dst_positive_scale_exists_next_maskentryinput))) + (((dst_negative_code_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)) * S ((dst_negative_code_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)) + ((dst_negative_scale_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)))) + ((((dst_negative_code_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)) * S ((dst_negative_code_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)) + ((dst_negative_scale_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput))) + (((dst_negative_code_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)) * S ((dst_negative_code_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)) + ((dst_negative_scale_exists_next_maskentryinput) + (dst_negative_scale_exists_next_maskentryinput)))))) /\ (((((exists ff_h_pvs_exists_next_maskentryinputpositive. ff_h_pvs_exists_next_maskentryinputpositive + S (dst_positive_exists_next_maskentryinput) = S ((S (dm_index_exists_next_mask)) * dst_positive_scale_exists_next_maskentryinput)) /\ exists ff_q_pvs_exists_next_maskentryinputpositive. dst_positive_code_exists_next_maskentryinput = ff_q_pvs_exists_next_maskentryinputpositive * S ((S (dm_index_exists_next_mask)) * dst_positive_scale_exists_next_maskentryinput) + (dst_positive_exists_next_maskentryinput))) /\ (((((exists ff_h_pvs_exists_next_maskentryinputnegative. ff_h_pvs_exists_next_maskentryinputnegative + S (dst_negative_exists_next_maskentryinput) = S ((S (dm_index_exists_next_mask)) * dst_negative_scale_exists_next_maskentryinput)) /\ exists ff_q_pvs_exists_next_maskentryinputnegative. dst_negative_code_exists_next_maskentryinput = ff_q_pvs_exists_next_maskentryinputnegative * S ((S (dm_index_exists_next_mask)) * dst_negative_scale_exists_next_maskentryinput) + (dst_negative_exists_next_maskentryinput))) /\ (exists ge_balance_positive_exists_next_maskentryinputvalue ge_balance_negative_exists_next_maskentryinputvalue. (((((dm_value_exists_next_mask) = 2 * (ge_balance_positive_exists_next_maskentryinputvalue) /\ (ge_balance_negative_exists_next_maskentryinputvalue) = 0) \/ exists ge_signed_half_exists_next_maskentryinputvaluedecode. (((dm_value_exists_next_mask) = 2 * ge_signed_half_exists_next_maskentryinputvaluedecode + 1 /\ (ge_balance_positive_exists_next_maskentryinputvalue) = 0) /\ (ge_balance_negative_exists_next_maskentryinputvalue) = S ge_signed_half_exists_next_maskentryinputvaluedecode))) /\ ((dst_positive_exists_next_maskentryinput) + ge_balance_negative_exists_next_maskentryinputvalue = (dst_negative_exists_next_maskentryinput) + ge_balance_positive_exists_next_maskentryinputvalue))))))))))))) \/ ((((dm_index_exists_next_mask)=0 \/ ~(exists pvs_factor_exists_next_maskentrynondivisor. (n) = (dm_index_exists_next_mask) * pvs_factor_exists_next_maskentrynondivisor)) /\ ((dm_value_exists_next_mask)=0))))))) /\ (forall dst_index_exists_previous_values dst_first_exists_previous_values dst_second_exists_previous_values. (exists pvs_gap_exists_previous_valuesbound. pvs_gap_exists_previous_valuesbound + S (dst_index_exists_previous_values) = (S l)) -> (exists dst_positive_code_exists_previous_valuesfirst dst_positive_scale_exists_previous_valuesfirst dst_negative_code_exists_previous_valuesfirst dst_negative_scale_exists_previous_valuesfirst dst_positive_exists_previous_valuesfirst dst_negative_exists_previous_valuesfirst. (((x) = (((((dst_positive_code_exists_previous_valuesfirst) + (dst_positive_scale_exists_previous_valuesfirst)) * S ((dst_positive_code_exists_previous_valuesfirst) + (dst_positive_scale_exists_previous_valuesfirst)) + ((dst_positive_scale_exists_previous_valuesfirst) + (dst_positive_scale_exists_previous_valuesfirst))) + (((dst_negative_code_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)) * S ((dst_negative_code_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)) + ((dst_negative_scale_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)))) * S ((((dst_positive_code_exists_previous_valuesfirst) + (dst_positive_scale_exists_previous_valuesfirst)) * S ((dst_positive_code_exists_previous_valuesfirst) + (dst_positive_scale_exists_previous_valuesfirst)) + ((dst_positive_scale_exists_previous_valuesfirst) + (dst_positive_scale_exists_previous_valuesfirst))) + (((dst_negative_code_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)) * S ((dst_negative_code_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)) + ((dst_negative_scale_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)))) + ((((dst_negative_code_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)) * S ((dst_negative_code_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)) + ((dst_negative_scale_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst))) + (((dst_negative_code_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)) * S ((dst_negative_code_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)) + ((dst_negative_scale_exists_previous_valuesfirst) + (dst_negative_scale_exists_previous_valuesfirst)))))) /\ (((((exists ff_h_pvs_exists_previous_valuesfirstpositive. ff_h_pvs_exists_previous_valuesfirstpositive + S (dst_positive_exists_previous_valuesfirst) = S ((S (dst_index_exists_previous_values)) * dst_positive_scale_exists_previous_valuesfirst)) /\ exists ff_q_pvs_exists_previous_valuesfirstpositive. dst_positive_code_exists_previous_valuesfirst = ff_q_pvs_exists_previous_valuesfirstpositive * S ((S (dst_index_exists_previous_values)) * dst_positive_scale_exists_previous_valuesfirst) + (dst_positive_exists_previous_valuesfirst))) /\ (((((exists ff_h_pvs_exists_previous_valuesfirstnegative. ff_h_pvs_exists_previous_valuesfirstnegative + S (dst_negative_exists_previous_valuesfirst) = S ((S (dst_index_exists_previous_values)) * dst_negative_scale_exists_previous_valuesfirst)) /\ exists ff_q_pvs_exists_previous_valuesfirstnegative. dst_negative_code_exists_previous_valuesfirst = ff_q_pvs_exists_previous_valuesfirstnegative * S ((S (dst_index_exists_previous_values)) * dst_negative_scale_exists_previous_valuesfirst) + (dst_negative_exists_previous_valuesfirst))) /\ (exists ge_balance_positive_exists_previous_valuesfirstvalue ge_balance_negative_exists_previous_valuesfirstvalue. (((((dst_first_exists_previous_values) = 2 * (ge_balance_positive_exists_previous_valuesfirstvalue) /\ (ge_balance_negative_exists_previous_valuesfirstvalue) = 0) \/ exists ge_signed_half_exists_previous_valuesfirstvaluedecode. (((dst_first_exists_previous_values) = 2 * ge_signed_half_exists_previous_valuesfirstvaluedecode + 1 /\ (ge_balance_positive_exists_previous_valuesfirstvalue) = 0) /\ (ge_balance_negative_exists_previous_valuesfirstvalue) = S ge_signed_half_exists_previous_valuesfirstvaluedecode))) /\ ((dst_positive_exists_previous_valuesfirst) + ge_balance_negative_exists_previous_valuesfirstvalue = (dst_negative_exists_previous_valuesfirst) + ge_balance_positive_exists_previous_valuesfirstvalue))))))))) -> (exists dst_positive_code_exists_previous_valuessecond dst_positive_scale_exists_previous_valuessecond dst_negative_code_exists_previous_valuessecond dst_negative_scale_exists_previous_valuessecond dst_positive_exists_previous_valuessecond dst_negative_exists_previous_valuessecond. (((G) = (((((dst_positive_code_exists_previous_valuessecond) + (dst_positive_scale_exists_previous_valuessecond)) * S ((dst_positive_code_exists_previous_valuessecond) + (dst_positive_scale_exists_previous_valuessecond)) + ((dst_positive_scale_exists_previous_valuessecond) + (dst_positive_scale_exists_previous_valuessecond))) + (((dst_negative_code_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)) * S ((dst_negative_code_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)) + ((dst_negative_scale_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)))) * S ((((dst_positive_code_exists_previous_valuessecond) + (dst_positive_scale_exists_previous_valuessecond)) * S ((dst_positive_code_exists_previous_valuessecond) + (dst_positive_scale_exists_previous_valuessecond)) + ((dst_positive_scale_exists_previous_valuessecond) + (dst_positive_scale_exists_previous_valuessecond))) + (((dst_negative_code_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)) * S ((dst_negative_code_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)) + ((dst_negative_scale_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)))) + ((((dst_negative_code_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)) * S ((dst_negative_code_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)) + ((dst_negative_scale_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond))) + (((dst_negative_code_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)) * S ((dst_negative_code_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)) + ((dst_negative_scale_exists_previous_valuessecond) + (dst_negative_scale_exists_previous_valuessecond)))))) /\ (((((exists ff_h_pvs_exists_previous_valuessecondpositive. ff_h_pvs_exists_previous_valuessecondpositive + S (dst_positive_exists_previous_valuessecond) = S ((S (dst_index_exists_previous_values)) * dst_positive_scale_exists_previous_valuessecond)) /\ exists ff_q_pvs_exists_previous_valuessecondpositive. dst_positive_code_exists_previous_valuessecond = ff_q_pvs_exists_previous_valuessecondpositive * S ((S (dst_index_exists_previous_values)) * dst_positive_scale_exists_previous_valuessecond) + (dst_positive_exists_previous_valuessecond))) /\ (((((exists ff_h_pvs_exists_previous_valuessecondnegative. ff_h_pvs_exists_previous_valuessecondnegative + S (dst_negative_exists_previous_valuessecond) = S ((S (dst_index_exists_previous_values)) * dst_negative_scale_exists_previous_valuessecond)) /\ exists ff_q_pvs_exists_previous_valuessecondnegative. dst_negative_code_exists_previous_valuessecond = ff_q_pvs_exists_previous_valuessecondnegative * S ((S (dst_index_exists_previous_values)) * dst_negative_scale_exists_previous_valuessecond) + (dst_negative_exists_previous_valuessecond))) /\ (exists ge_balance_positive_exists_previous_valuessecondvalue ge_balance_negative_exists_previous_valuessecondvalue. (((((dst_second_exists_previous_values) = 2 * (ge_balance_positive_exists_previous_valuessecondvalue) /\ (ge_balance_negative_exists_previous_valuessecondvalue) = 0) \/ exists ge_signed_half_exists_previous_valuessecondvaluedecode. (((dst_second_exists_previous_values) = 2 * ge_signed_half_exists_previous_valuessecondvaluedecode + 1 /\ (ge_balance_positive_exists_previous_valuessecondvalue) = 0) /\ (ge_balance_negative_exists_previous_valuessecondvalue) = S ge_signed_half_exists_previous_valuessecondvaluedecode))) /\ ((dst_positive_exists_previous_valuessecond) + ge_balance_negative_exists_previous_valuessecondvalue = (dst_negative_exists_previous_valuessecond) + ge_balance_positive_exists_previous_valuessecondvalue))))))))) -> dst_first_exists_previous_values = dst_second_exists_previous_values)
  45. 0045specialize divisor_mask_prefix_append (F)
  46. 0046specialize divisor_mask_prefix_append (n)
  47. 0047specialize divisor_mask_prefix_append (l)
  48. 0048specialize divisor_mask_prefix_append (x)
  49. 0049specialize divisor_mask_prefix_append (x1)
  50. 0050apply divisor_mask_prefix_append
  51. 0051exact hprev_witness
  52. 0052exact hz_witness
  53. 0053cases hnext
  54. 0054cases hnext_witness
  55. 0055exists x2
  56. 0056exact hnext_witness_left