PA0046 · theorem

pow_exists

Stable checked-use theorem · independently closed

Every base and exponent have a relational finite-product power.

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.

Statement with defined notation

∀ a. ∀ e. ∃ n. Pow(a,e,n)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0020 Pow

1 occurrences

In local proof propositions

PD0019 Repeat

1 occurrences

Exact expanded native-PA statement
forall a e. exists n. (exists ff_b_x ff_c_x. ((forall ff_i_x_repeat. (exists ff_lt_x_repeat_bound. ff_lt_x_repeat_bound + S ff_i_x_repeat = e) -> (((exists ff_h_x_repeat_decoded. ff_h_x_repeat_decoded + S (a) = S ((S (ff_i_x_repeat)) * ff_c_x)) /\ exists ff_q_x_repeat_decoded. ff_b_x = ff_q_x_repeat_decoded * S ((S (ff_i_x_repeat)) * ff_c_x) + (a)))) /\ (exists ff_u_x_product ff_v_x_product. ((((exists ff_h_x_product_start. ff_h_x_product_start + S (1) = S ((S (0)) * ff_v_x_product)) /\ exists ff_q_x_product_start. ff_u_x_product = ff_q_x_product_start * S ((S (0)) * ff_v_x_product) + (1))) /\ ((((exists ff_h_x_product_terminal. ff_h_x_product_terminal + S (n) = S ((S (e)) * ff_v_x_product)) /\ exists ff_q_x_product_terminal. ff_u_x_product = ff_q_x_product_terminal * S ((S (e)) * ff_v_x_product) + (n))) /\ forall ff_i_x_product. (exists ff_lt_x_product_bound. ff_lt_x_product_bound + S ff_i_x_product = e) -> exists ff_p_x_product ff_r_x_product ff_s_x_product. ((((exists ff_h_x_product_factor. ff_h_x_product_factor + S (ff_p_x_product) = S ((S (ff_i_x_product)) * ff_c_x)) /\ exists ff_q_x_product_factor. ff_b_x = ff_q_x_product_factor * S ((S (ff_i_x_product)) * ff_c_x) + (ff_p_x_product))) /\ ((((exists ff_h_x_product_partial. ff_h_x_product_partial + S (ff_r_x_product) = S ((S (ff_i_x_product)) * ff_v_x_product)) /\ exists ff_q_x_product_partial. ff_u_x_product = ff_q_x_product_partial * S ((S (ff_i_x_product)) * ff_v_x_product) + (ff_r_x_product))) /\ ((((exists ff_h_x_product_successor. ff_h_x_product_successor + S (ff_s_x_product) = S ((S (S ff_i_x_product)) * ff_v_x_product)) /\ exists ff_q_x_product_successor. ff_u_x_product = ff_q_x_product_successor * S ((S (S ff_i_x_product)) * ff_v_x_product) + (ff_s_x_product))) /\ ff_s_x_product = ff_r_x_product * ff_p_x_product))))))))

Proof neighborhood

Direct theorem prerequisites

PA0045 beta_repeat_exists PA003X beta_product_exists

Direct theorem dependents

PA005Y pow_mul_exp PA007J beta_sign_factor_product_power_exists PA0085 gauss_lemma_power_congruence_exists PA0088 pow_congruent_base_witness PA008M quadratic_residue_half_power_mod_one

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

22 script commands · 10 reading checkpoints · 1 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)
beta repeat existsbeta product exists
01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro e
02Establish hrepeatL3–6

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

  1. L3
    have hrepeat : ∃ b. ∃ c. Repeat(b,c,a,e)Definitions: Repeat(b,c,a,e)Original native command in the exact edition
  2. L4
    specialize beta_repeat_exists a
  3. L5
    specialize beta_repeat_exists e
  4. L6
    exact beta_repeat_exists
03Separate the logical casesL7–8

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

  1. L7
    cases hrepeat
  2. L8
    cases hrepeat_witness
04Use earlier factsL9–11

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

  1. L9
    specialize beta_product_exists x
  2. L10
    specialize beta_product_exists x1
  3. L11
    specialize beta_product_exists e
05Separate the logical casesL12–14

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

  1. L12
    cases beta_product_exists
  2. L13
    cases beta_product_exists_witness
  3. L14
    cases beta_product_exists_witness_witness
06Construct an explicit witnessL15–17

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

  1. L15
    exists x2
  2. L16
    exists x
  3. L17
    exists x1
07Separate the logical casesL18–18

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

  1. L18
    split
08Use earlier factsL19–19

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

  1. L19
    exact hrepeat_witness_witness
09Construct an explicit witnessL20–21

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

  1. L20
    exists x3
  2. L21
    exists x4
10Use earlier factsL22–22

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

  1. L22
    exact beta_product_exists_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro a
  2. 0002intro e
  3. 0003have hrepeat : ∃ b. ∃ c. Repeat(b,c,a,e)
    Exact native replay linehave hrepeat : exists b c. (forall i. (exists h. h + S i = e) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a))
  4. 0004specialize beta_repeat_exists a
  5. 0005specialize beta_repeat_exists e
  6. 0006exact beta_repeat_exists
  7. 0007cases hrepeat
  8. 0008cases hrepeat_witness
  9. 0009specialize beta_product_exists x
  10. 0010specialize beta_product_exists x1
  11. 0011specialize beta_product_exists e
  12. 0012cases beta_product_exists
  13. 0013cases beta_product_exists_witness
  14. 0014cases beta_product_exists_witness_witness
  15. 0015exists x2
  16. 0016exists x
  17. 0017exists x1
  18. 0018split
  19. 0019exact hrepeat_witness_witness
  20. 0020exists x3
  21. 0021exists x4
  22. 0022exact beta_product_exists_witness_witness_witness