PA005U · theorem

pow_one

Stable checked-use theorem · independently closed

The relational first power of a natural is the natural itself.

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. e = 1 → Pow(a,e,n) → n = a

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

none

0 occurrences

Exact expanded native-PA statement
forall a e n. e = 1 -> (exists ff_b_one ff_c_one. ((forall ff_i_one_repeat. (exists ff_lt_one_repeat_bound. ff_lt_one_repeat_bound + S ff_i_one_repeat = e) -> (((exists ff_h_one_repeat_decoded. ff_h_one_repeat_decoded + S (a) = S ((S (ff_i_one_repeat)) * ff_c_one)) /\ exists ff_q_one_repeat_decoded. ff_b_one = ff_q_one_repeat_decoded * S ((S (ff_i_one_repeat)) * ff_c_one) + (a)))) /\ (exists ff_u_one_product ff_v_one_product. ((((exists ff_h_one_product_start. ff_h_one_product_start + S (1) = S ((S (0)) * ff_v_one_product)) /\ exists ff_q_one_product_start. ff_u_one_product = ff_q_one_product_start * S ((S (0)) * ff_v_one_product) + (1))) /\ ((((exists ff_h_one_product_terminal. ff_h_one_product_terminal + S (n) = S ((S (e)) * ff_v_one_product)) /\ exists ff_q_one_product_terminal. ff_u_one_product = ff_q_one_product_terminal * S ((S (e)) * ff_v_one_product) + (n))) /\ forall ff_i_one_product. (exists ff_lt_one_product_bound. ff_lt_one_product_bound + S ff_i_one_product = e) -> exists ff_p_one_product ff_r_one_product ff_s_one_product. ((((exists ff_h_one_product_factor. ff_h_one_product_factor + S (ff_p_one_product) = S ((S (ff_i_one_product)) * ff_c_one)) /\ exists ff_q_one_product_factor. ff_b_one = ff_q_one_product_factor * S ((S (ff_i_one_product)) * ff_c_one) + (ff_p_one_product))) /\ ((((exists ff_h_one_product_partial. ff_h_one_product_partial + S (ff_r_one_product) = S ((S (ff_i_one_product)) * ff_v_one_product)) /\ exists ff_q_one_product_partial. ff_u_one_product = ff_q_one_product_partial * S ((S (ff_i_one_product)) * ff_v_one_product) + (ff_r_one_product))) /\ ((((exists ff_h_one_product_successor. ff_h_one_product_successor + S (ff_s_one_product) = S ((S (S ff_i_one_product)) * ff_v_one_product)) /\ exists ff_q_one_product_successor. ff_u_one_product = ff_q_one_product_successor * S ((S (S ff_i_one_product)) * ff_v_one_product) + (ff_s_one_product))) /\ ff_s_one_product = ff_r_one_product * ff_p_one_product)))))))) -> n = a

Proof neighborhood

Direct theorem prerequisites

PA005T pow_one_from_zero_successor

Direct theorem dependents

PA005V pow_two_from_one_successor

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

13 script commands · 4 reading checkpoints · 0 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 (1)
pow one from zero successor
01Fix variables and assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro e
  3. L3
    intro n
  4. L4
    intro he
  5. L5
    intro hpow
02Use earlier factsL6–10

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

  1. L6
    specialize pow_one_from_zero_successor a
  2. L7
    specialize pow_one_from_zero_successor 0
  3. L8
    specialize pow_one_from_zero_successor e
  4. L9
    specialize pow_one_from_zero_successor n
  5. L10
    apply pow_one_from_zero_successor
03Calculate and transport equalitiesL11–11

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L11
    refl
04Use earlier factsL12–13

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

  1. L12
    exact he
  2. L13
    exact hpow

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro a
  2. 0002intro e
  3. 0003intro n
  4. 0004intro he
  5. 0005intro hpow
  6. 0006specialize pow_one_from_zero_successor a
  7. 0007specialize pow_one_from_zero_successor 0
  8. 0008specialize pow_one_from_zero_successor e
  9. 0009specialize pow_one_from_zero_successor n
  10. 0010apply pow_one_from_zero_successor
  11. 0011refl
  12. 0012exact he
  13. 0013exact hpow