SS000E

divisor_signed_sum_empty_exists

The empty sum is constructed as an actual two-trace signed fold and only then identified with zero.

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

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

These are actual signed-table and finite-sum foundations. Equality compares represented signed values, not arbitrary encodings. MatrixMinorFourCode is reused solely as generic nested pairing, without a matrix hypothesis. Full finite signed G007 is established separately in the Möbius-inversion family.

Exact theorem in conservative defined notation

∀ F. ∀ pb. ∀ pc. ∀ nb. ∀ nc. MatrixMinorFourCode(F,pb,pc,nb,nc) → SignedPrefixSum(F,0,0)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

MatrixMinorFourCode(z,up,us,un,ut) · 1SignedPrefixSum(F,l,z) · 2

Actual proof prerequisites

divisor_signed_sum_exists_from_componentsdivisor_signed_sum_empty_value
Original expanded first-order statement 
forall F pb pc nb nc. ((F) = (((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) * S ((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) + ((((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))))) -> (exists dst_positive_code_empty_result dst_positive_scale_empty_result dst_negative_code_empty_result dst_negative_scale_empty_result dst_positive_sum_empty_result dst_negative_sum_empty_result. (((F) = (((((dst_positive_code_empty_result) + (dst_positive_scale_empty_result)) * S ((dst_positive_code_empty_result) + (dst_positive_scale_empty_result)) + ((dst_positive_scale_empty_result) + (dst_positive_scale_empty_result))) + (((dst_negative_code_empty_result) + (dst_negative_scale_empty_result)) * S ((dst_negative_code_empty_result) + (dst_negative_scale_empty_result)) + ((dst_negative_scale_empty_result) + (dst_negative_scale_empty_result)))) * S ((((dst_positive_code_empty_result) + (dst_positive_scale_empty_result)) * S ((dst_positive_code_empty_result) + (dst_positive_scale_empty_result)) + ((dst_positive_scale_empty_result) + (dst_positive_scale_empty_result))) + (((dst_negative_code_empty_result) + (dst_negative_scale_empty_result)) * S ((dst_negative_code_empty_result) + (dst_negative_scale_empty_result)) + ((dst_negative_scale_empty_result) + (dst_negative_scale_empty_result)))) + ((((dst_negative_code_empty_result) + (dst_negative_scale_empty_result)) * S ((dst_negative_code_empty_result) + (dst_negative_scale_empty_result)) + ((dst_negative_scale_empty_result) + (dst_negative_scale_empty_result))) + (((dst_negative_code_empty_result) + (dst_negative_scale_empty_result)) * S ((dst_negative_code_empty_result) + (dst_negative_scale_empty_result)) + ((dst_negative_scale_empty_result) + (dst_negative_scale_empty_result)))))) /\ (((exists fs_u_dst_empty_resultpositive fs_v_dst_empty_resultpositive. ((((exists fs_h_dst_empty_resultpositive_body_start. fs_h_dst_empty_resultpositive_body_start + S (0) = S ((S (0)) * fs_v_dst_empty_resultpositive)) /\ exists fs_q_dst_empty_resultpositive_body_start. fs_u_dst_empty_resultpositive = fs_q_dst_empty_resultpositive_body_start * S ((S (0)) * fs_v_dst_empty_resultpositive) + (0))) /\ ((((exists fs_h_dst_empty_resultpositive_body_terminal. fs_h_dst_empty_resultpositive_body_terminal + S (dst_positive_sum_empty_result) = S ((S (0)) * fs_v_dst_empty_resultpositive)) /\ exists fs_q_dst_empty_resultpositive_body_terminal. fs_u_dst_empty_resultpositive = fs_q_dst_empty_resultpositive_body_terminal * S ((S (0)) * fs_v_dst_empty_resultpositive) + (dst_positive_sum_empty_result))) /\ forall fs_i_dst_empty_resultpositive_body_steps. (exists fs_lt_dst_empty_resultpositive_body_steps_bound. fs_lt_dst_empty_resultpositive_body_steps_bound + S fs_i_dst_empty_resultpositive_body_steps = 0) -> exists fs_a_dst_empty_resultpositive_body_steps fs_r_dst_empty_resultpositive_body_steps fs_s_dst_empty_resultpositive_body_steps. ((((exists fs_h_dst_empty_resultpositive_body_steps_summand. fs_h_dst_empty_resultpositive_body_steps_summand + S (fs_a_dst_empty_resultpositive_body_steps) = S ((S (fs_i_dst_empty_resultpositive_body_steps)) * dst_positive_scale_empty_result)) /\ exists fs_q_dst_empty_resultpositive_body_steps_summand. dst_positive_code_empty_result = fs_q_dst_empty_resultpositive_body_steps_summand * S ((S (fs_i_dst_empty_resultpositive_body_steps)) * dst_positive_scale_empty_result) + (fs_a_dst_empty_resultpositive_body_steps))) /\ ((((exists fs_h_dst_empty_resultpositive_body_steps_partial. fs_h_dst_empty_resultpositive_body_steps_partial + S (fs_r_dst_empty_resultpositive_body_steps) = S ((S (fs_i_dst_empty_resultpositive_body_steps)) * fs_v_dst_empty_resultpositive)) /\ exists fs_q_dst_empty_resultpositive_body_steps_partial. fs_u_dst_empty_resultpositive = fs_q_dst_empty_resultpositive_body_steps_partial * S ((S (fs_i_dst_empty_resultpositive_body_steps)) * fs_v_dst_empty_resultpositive) + (fs_r_dst_empty_resultpositive_body_steps))) /\ ((((exists fs_h_dst_empty_resultpositive_body_steps_successor. fs_h_dst_empty_resultpositive_body_steps_successor + S (fs_s_dst_empty_resultpositive_body_steps) = S ((S (S fs_i_dst_empty_resultpositive_body_steps)) * fs_v_dst_empty_resultpositive)) /\ exists fs_q_dst_empty_resultpositive_body_steps_successor. fs_u_dst_empty_resultpositive = fs_q_dst_empty_resultpositive_body_steps_successor * S ((S (S fs_i_dst_empty_resultpositive_body_steps)) * fs_v_dst_empty_resultpositive) + (fs_s_dst_empty_resultpositive_body_steps))) /\ fs_s_dst_empty_resultpositive_body_steps = fs_r_dst_empty_resultpositive_body_steps + fs_a_dst_empty_resultpositive_body_steps)))))) /\ (((exists fs_u_dst_empty_resultnegative fs_v_dst_empty_resultnegative. ((((exists fs_h_dst_empty_resultnegative_body_start. fs_h_dst_empty_resultnegative_body_start + S (0) = S ((S (0)) * fs_v_dst_empty_resultnegative)) /\ exists fs_q_dst_empty_resultnegative_body_start. fs_u_dst_empty_resultnegative = fs_q_dst_empty_resultnegative_body_start * S ((S (0)) * fs_v_dst_empty_resultnegative) + (0))) /\ ((((exists fs_h_dst_empty_resultnegative_body_terminal. fs_h_dst_empty_resultnegative_body_terminal + S (dst_negative_sum_empty_result) = S ((S (0)) * fs_v_dst_empty_resultnegative)) /\ exists fs_q_dst_empty_resultnegative_body_terminal. fs_u_dst_empty_resultnegative = fs_q_dst_empty_resultnegative_body_terminal * S ((S (0)) * fs_v_dst_empty_resultnegative) + (dst_negative_sum_empty_result))) /\ forall fs_i_dst_empty_resultnegative_body_steps. (exists fs_lt_dst_empty_resultnegative_body_steps_bound. fs_lt_dst_empty_resultnegative_body_steps_bound + S fs_i_dst_empty_resultnegative_body_steps = 0) -> exists fs_a_dst_empty_resultnegative_body_steps fs_r_dst_empty_resultnegative_body_steps fs_s_dst_empty_resultnegative_body_steps. ((((exists fs_h_dst_empty_resultnegative_body_steps_summand. fs_h_dst_empty_resultnegative_body_steps_summand + S (fs_a_dst_empty_resultnegative_body_steps) = S ((S (fs_i_dst_empty_resultnegative_body_steps)) * dst_negative_scale_empty_result)) /\ exists fs_q_dst_empty_resultnegative_body_steps_summand. dst_negative_code_empty_result = fs_q_dst_empty_resultnegative_body_steps_summand * S ((S (fs_i_dst_empty_resultnegative_body_steps)) * dst_negative_scale_empty_result) + (fs_a_dst_empty_resultnegative_body_steps))) /\ ((((exists fs_h_dst_empty_resultnegative_body_steps_partial. fs_h_dst_empty_resultnegative_body_steps_partial + S (fs_r_dst_empty_resultnegative_body_steps) = S ((S (fs_i_dst_empty_resultnegative_body_steps)) * fs_v_dst_empty_resultnegative)) /\ exists fs_q_dst_empty_resultnegative_body_steps_partial. fs_u_dst_empty_resultnegative = fs_q_dst_empty_resultnegative_body_steps_partial * S ((S (fs_i_dst_empty_resultnegative_body_steps)) * fs_v_dst_empty_resultnegative) + (fs_r_dst_empty_resultnegative_body_steps))) /\ ((((exists fs_h_dst_empty_resultnegative_body_steps_successor. fs_h_dst_empty_resultnegative_body_steps_successor + S (fs_s_dst_empty_resultnegative_body_steps) = S ((S (S fs_i_dst_empty_resultnegative_body_steps)) * fs_v_dst_empty_resultnegative)) /\ exists fs_q_dst_empty_resultnegative_body_steps_successor. fs_u_dst_empty_resultnegative = fs_q_dst_empty_resultnegative_body_steps_successor * S ((S (S fs_i_dst_empty_resultnegative_body_steps)) * fs_v_dst_empty_resultnegative) + (fs_s_dst_empty_resultnegative_body_steps))) /\ fs_s_dst_empty_resultnegative_body_steps = fs_r_dst_empty_resultnegative_body_steps + fs_a_dst_empty_resultnegative_body_steps)))))) /\ (exists ge_balance_positive_empty_resultresult ge_balance_negative_empty_resultresult. (((((0) = 2 * (ge_balance_positive_empty_resultresult) /\ (ge_balance_negative_empty_resultresult) = 0) \/ exists ge_signed_half_empty_resultresultdecode. (((0) = 2 * ge_signed_half_empty_resultresultdecode + 1 /\ (ge_balance_positive_empty_resultresult) = 0) /\ (ge_balance_negative_empty_resultresult) = S ge_signed_half_empty_resultresultdecode))) /\ ((dst_positive_sum_empty_result) + ge_balance_negative_empty_resultresult = (dst_negative_sum_empty_result) + ge_balance_positive_empty_resultresult)))))))))

Complete tactic proof in conservative notation

All 24 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

24 script commands · 4 reading checkpoints · 2 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (2)
divisor signed sum exists from componentsdivisor signed sum empty value
01Fix variables and assumptionsL1–6

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

  1. L1
    intro F
  2. L2
    intro pb
  3. L3
    intro pc
  4. L4
    intro nb
  5. L5
    intro nc
  6. L6
    intro hrep
02Establish hzL7–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply divisor signed sum exists from components.

  1. L7
    have hz : ∃ z. SignedPrefixSum(F,0,z)Definitions: SignedPrefixSum(F,0,z)Original native command in the exact edition
  2. L8
    specialize divisor_signed_sum_exists_from_components (F)
  3. L9
    specialize divisor_signed_sum_exists_from_components (pb)
  4. L10
    specialize divisor_signed_sum_exists_from_components (pc)
  5. L11
    specialize divisor_signed_sum_exists_from_components (nb)
  6. L12
    specialize divisor_signed_sum_exists_from_components (nc)
  7. L13
    specialize divisor_signed_sum_exists_from_components (0)
  8. L14
    apply divisor_signed_sum_exists_from_components
  9. L15
    exact hrep
03Separate the logical casesL16–16

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

  1. L16
    cases hz
04Establish heqL17–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply divisor signed sum empty value.

  1. L17
    have heq : x = 0
  2. L18
    specialize divisor_signed_sum_empty_value (F)
  3. L19
    specialize divisor_signed_sum_empty_value (x)
  4. L20
    apply divisor_signed_sum_empty_value
  5. L21
    exact hz_witness
  6. L22
    rewrite heq at hz_witness
  7. L23
    rewrite heq at hz_witness
  8. L24
    exact hz_witness

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro F
  2. 0002intro pb
  3. 0003intro pc
  4. 0004intro nb
  5. 0005intro nc
  6. 0006intro hrep
  7. 0007have hz : ∃ z. SignedPrefixSum(F,0,z)
  8. 0008specialize divisor_signed_sum_exists_from_components (F)
  9. 0009specialize divisor_signed_sum_exists_from_components (pb)
  10. 0010specialize divisor_signed_sum_exists_from_components (pc)
  11. 0011specialize divisor_signed_sum_exists_from_components (nb)
  12. 0012specialize divisor_signed_sum_exists_from_components (nc)
  13. 0013specialize divisor_signed_sum_exists_from_components (0)
  14. 0014apply divisor_signed_sum_exists_from_components
  15. 0015exact hrep
  16. 0016cases hz
  17. 0017have heq : x = 0
  18. 0018specialize divisor_signed_sum_empty_value (F)
  19. 0019specialize divisor_signed_sum_empty_value (x)
  20. 0020apply divisor_signed_sum_empty_value
  21. 0021exact hz_witness
  22. 0022rewrite heq at hz_witness
  23. 0023rewrite heq at hz_witness
  24. 0024exact hz_witness