PA0066 · theorem

factorial_zero

Stable checked-use theorem · independently closed

The relational factorial of zero is one.

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

∀ n. ∀ z. n = 0 → Factorial(n,z) → z = 1

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

PD0023 Factorial

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall n z. n = 0 -> (exists ff_b_zero ff_c_zero. ((forall ff_i_zero_range. (exists ff_lt_zero_range_bound. ff_lt_zero_range_bound + S ff_i_zero_range = n) -> (((exists ff_h_zero_range_decoded. ff_h_zero_range_decoded + S (1 + ff_i_zero_range) = S ((S (ff_i_zero_range)) * ff_c_zero)) /\ exists ff_q_zero_range_decoded. ff_b_zero = ff_q_zero_range_decoded * S ((S (ff_i_zero_range)) * ff_c_zero) + (1 + ff_i_zero_range)))) /\ (exists ff_u_zero_product ff_v_zero_product. ((((exists ff_h_zero_product_start. ff_h_zero_product_start + S (1) = S ((S (0)) * ff_v_zero_product)) /\ exists ff_q_zero_product_start. ff_u_zero_product = ff_q_zero_product_start * S ((S (0)) * ff_v_zero_product) + (1))) /\ ((((exists ff_h_zero_product_terminal. ff_h_zero_product_terminal + S (z) = S ((S (n)) * ff_v_zero_product)) /\ exists ff_q_zero_product_terminal. ff_u_zero_product = ff_q_zero_product_terminal * S ((S (n)) * ff_v_zero_product) + (z))) /\ forall ff_i_zero_product. (exists ff_lt_zero_product_bound. ff_lt_zero_product_bound + S ff_i_zero_product = n) -> exists ff_p_zero_product ff_r_zero_product ff_s_zero_product. ((((exists ff_h_zero_product_factor. ff_h_zero_product_factor + S (ff_p_zero_product) = S ((S (ff_i_zero_product)) * ff_c_zero)) /\ exists ff_q_zero_product_factor. ff_b_zero = ff_q_zero_product_factor * S ((S (ff_i_zero_product)) * ff_c_zero) + (ff_p_zero_product))) /\ ((((exists ff_h_zero_product_partial. ff_h_zero_product_partial + S (ff_r_zero_product) = S ((S (ff_i_zero_product)) * ff_v_zero_product)) /\ exists ff_q_zero_product_partial. ff_u_zero_product = ff_q_zero_product_partial * S ((S (ff_i_zero_product)) * ff_v_zero_product) + (ff_r_zero_product))) /\ ((((exists ff_h_zero_product_successor. ff_h_zero_product_successor + S (ff_s_zero_product) = S ((S (S ff_i_zero_product)) * ff_v_zero_product)) /\ exists ff_q_zero_product_successor. ff_u_zero_product = ff_q_zero_product_successor * S ((S (S ff_i_zero_product)) * ff_v_zero_product) + (ff_s_zero_product))) /\ ff_s_zero_product = ff_r_zero_product * ff_p_zero_product)))))))) -> z = 1

Proof neighborhood

Direct theorem prerequisites

PA0049 beta_product_zero

Direct theorem dependents

PA00A3 factorial_one_value

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

16 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)
beta product zero
01Fix variables and assumptionsL1–4

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

  1. L1
    intro n
  2. L2
    intro z
  3. L3
    intro hn
  4. L4
    intro hfactorial
02Calculate and transport equalitiesL5–8

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

  1. L5
    rewrite hn at hfactorial
  2. L6
    rewrite hn at hfactorial
  3. L7
    rewrite hn at hfactorial
  4. L8
    rewrite hn at hfactorial
03Separate the logical casesL9–11

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

  1. L9
    cases hfactorial
  2. L10
    cases hfactorial_witness
  3. L11
    cases hfactorial_witness_witness
04Use earlier factsL12–16

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

  1. L12
    specialize beta_product_zero x
  2. L13
    specialize beta_product_zero x1
  3. L14
    specialize beta_product_zero z
  4. L15
    apply beta_product_zero
  5. L16
    exact hfactorial_witness_witness_right

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro n
  2. 0002intro z
  3. 0003intro hn
  4. 0004intro hfactorial
  5. 0005rewrite hn at hfactorial
  6. 0006rewrite hn at hfactorial
  7. 0007rewrite hn at hfactorial
  8. 0008rewrite hn at hfactorial
  9. 0009cases hfactorial
  10. 0010cases hfactorial_witness
  11. 0011cases hfactorial_witness_witness
  12. 0012specialize beta_product_zero x
  13. 0013specialize beta_product_zero x1
  14. 0014specialize beta_product_zero z
  15. 0015apply beta_product_zero
  16. 0016exact hfactorial_witness_witness_right