PA003H

beta_sum_exists

Stable checked-use theorem · independently closed

Every decoded beta prefix has a relational finite sum.

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 original first-admission records.

Exact expanded PA statement

forall b c l. exists n. (exists ff_u_x ff_v_x. ((((exists ff_h_x_start. ff_h_x_start + S (0) = S ((S (0)) * ff_v_x)) /\ exists ff_q_x_start. ff_u_x = ff_q_x_start * S ((S (0)) * ff_v_x) + (0))) /\ ((((exists ff_h_x_terminal. ff_h_x_terminal + S (n) = S ((S (l)) * ff_v_x)) /\ exists ff_q_x_terminal. ff_u_x = ff_q_x_terminal * S ((S (l)) * ff_v_x) + (n))) /\ forall ff_i_x. (exists ff_lt_x_bound. ff_lt_x_bound + S ff_i_x = l) -> exists ff_a_x ff_r_x ff_s_x. ((((exists ff_h_x_summand. ff_h_x_summand + S (ff_a_x) = S ((S (ff_i_x)) * c)) /\ exists ff_q_x_summand. b = ff_q_x_summand * S ((S (ff_i_x)) * c) + (ff_a_x))) /\ ((((exists ff_h_x_partial. ff_h_x_partial + S (ff_r_x) = S ((S (ff_i_x)) * ff_v_x)) /\ exists ff_q_x_partial. ff_u_x = ff_q_x_partial * S ((S (ff_i_x)) * ff_v_x) + (ff_r_x))) /\ ((((exists ff_h_x_successor. ff_h_x_successor + S (ff_s_x) = S ((S (S ff_i_x)) * ff_v_x)) /\ exists ff_q_x_successor. ff_u_x = ff_q_x_successor * S ((S (S ff_i_x)) * ff_v_x) + (ff_s_x))) /\ ff_s_x = ff_r_x + ff_a_x))))))

Structural proof guide

Generated structural guide

Every decoded beta prefix has a relational finite sum.

Use the direct prerequisites beta_prefix_sum_trace_exists, beta_at_exists as previously established PA formulas.

The proof proceeds by case analysis (4), intermediate claims (2).

Referenced ingredients

PA003G beta_prefix_sum_trace_exists PA0029 beta_at_exists

Proof neighborhood

Direct dependencies

PA003G beta_prefix_sum_trace_exists PA0029 beta_at_exists

Direct dependents

PA003I bit_count_exists PA00C1 prime_scaled_half_quotient_sum_exists PA00CX beta_sum_permutation_invariant PA00D6 gauss_eisenstein_sign_count_mod_quotient_sum PA00DM distinct_odd_prime_half_rectangle_total_exists PA00EJ eisenstein_transposed_column_count_total_exists PA00EL beta_repeat_sum_exists_exact PA00FC eisenstein_fubini_universal PA00FF distinct_odd_prime_eisenstein_quotient_sum_identity

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

Read the argument

Proof checkpoints

25 script commands · 10 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.

Named ingredients (2)
beta prefix sum trace existsbeta at exists

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 assumptionsL1–3

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
02Establish htraceL4–8

Establish this local claim before using it. It is not an additional assumption.

  1. L4
    have htrace : ∃ fs_u_exists_trace. ∃ fs_v_exists_trace. BetaAt(fs_u_exists_trace,fs_v_exists_trace,0,0) ∧ (∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ n. BetaAt(b,c,x,y) ∧ (BetaAt(fs_u_exists_trace,fs_v_exists_trace,x,z) ∧ (BetaAt(fs_u_exists_trace,fs_v_exists_trace,S x,n) ∧ n = z + y)))Definitions: LtBetaAt
  2. L5
    specialize beta_prefix_sum_trace_exists b
  3. L6
    specialize beta_prefix_sum_trace_exists c
  4. L7
    specialize beta_prefix_sum_trace_exists l
  5. L8
    exact beta_prefix_sum_trace_exists
03Separate the logical casesL9–11

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

  1. L9
    cases htrace
  2. L10
    cases htrace_witness
  3. L11
    cases htrace_witness_witness
04Establish hterminalL12–16

Establish this local claim before using it. It is not an additional assumption.

  1. L12
    have hterminal : exists n. ((exists fs_h_sum_terminal. fs_h_sum_terminal + S (n) = S ((S (l)) * x1)) /\ exists fs_q_sum_terminal. x = fs_q_sum_terminal * S ((S (l)) * x1) + (n))
  2. L13
    specialize beta_at_exists x
  3. L14
    specialize beta_at_exists x1
  4. L15
    specialize beta_at_exists l
  5. L16
    exact beta_at_exists
05Separate the logical casesL17–17

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

  1. L17
    cases hterminal
06Construct an explicit witnessL18–20

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

  1. L18
    exists x2
  2. L19
    exists x
  3. L20
    exists x1
07Separate the logical casesL21–21

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

  1. L21
    split
08Use earlier factsL22–22

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

  1. L22
    exact htrace_witness_witness_left
09Separate the logical casesL23–23

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

  1. L23
    split
10Use earlier factsL24–25

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

  1. L24
    exact hterminal_witness
  2. L25
    exact htrace_witness_witness_right

Library-wide reading audit

Original exact command ledger · 25 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004have htrace : exists fs_u_exists_trace fs_v_exists_trace. ((((exists fs_h_exists_trace_start. fs_h_exists_trace_start + S (0) = S ((S (0)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_start. fs_u_exists_trace = fs_q_exists_trace_start * S ((S (0)) * fs_v_exists_trace) + (0))) /\ forall fs_i_exists_trace_steps. (exists fs_lt_exists_trace_steps_bound. fs_lt_exists_trace_steps_bound + S fs_i_exists_trace_steps = l) -> exists fs_a_exists_trace_steps fs_r_exists_trace_steps fs_s_exists_trace_steps. ((((exists fs_h_exists_trace_steps_summand. fs_h_exists_trace_steps_summand + S (fs_a_exists_trace_steps) = S ((S (fs_i_exists_trace_steps)) * c)) /\ exists fs_q_exists_trace_steps_summand. b = fs_q_exists_trace_steps_summand * S ((S (fs_i_exists_trace_steps)) * c) + (fs_a_exists_trace_steps))) /\ ((((exists fs_h_exists_trace_steps_partial. fs_h_exists_trace_steps_partial + S (fs_r_exists_trace_steps) = S ((S (fs_i_exists_trace_steps)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_steps_partial. fs_u_exists_trace = fs_q_exists_trace_steps_partial * S ((S (fs_i_exists_trace_steps)) * fs_v_exists_trace) + (fs_r_exists_trace_steps))) /\ ((((exists fs_h_exists_trace_steps_successor. fs_h_exists_trace_steps_successor + S (fs_s_exists_trace_steps) = S ((S (S fs_i_exists_trace_steps)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_steps_successor. fs_u_exists_trace = fs_q_exists_trace_steps_successor * S ((S (S fs_i_exists_trace_steps)) * fs_v_exists_trace) + (fs_s_exists_trace_steps))) /\ fs_s_exists_trace_steps = fs_r_exists_trace_steps + fs_a_exists_trace_steps))))
  5. 0005specialize beta_prefix_sum_trace_exists b
  6. 0006specialize beta_prefix_sum_trace_exists c
  7. 0007specialize beta_prefix_sum_trace_exists l
  8. 0008exact beta_prefix_sum_trace_exists
  9. 0009cases htrace
  10. 0010cases htrace_witness
  11. 0011cases htrace_witness_witness
  12. 0012have hterminal : exists n. ((exists fs_h_sum_terminal. fs_h_sum_terminal + S (n) = S ((S (l)) * x1)) /\ exists fs_q_sum_terminal. x = fs_q_sum_terminal * S ((S (l)) * x1) + (n))
  13. 0013specialize beta_at_exists x
  14. 0014specialize beta_at_exists x1
  15. 0015specialize beta_at_exists l
  16. 0016exact beta_at_exists
  17. 0017cases hterminal
  18. 0018exists x2
  19. 0019exists x
  20. 0020exists x1
  21. 0021split
  22. 0022exact htrace_witness_witness_left
  23. 0023split
  24. 0024exact hterminal_witness
  25. 0025exact htrace_witness_witness_right