PA00BO

double_predecessor_ne_one

Alpha v34 checked-use theorem · independently closed; not Stable

A doubled predecessor cannot equal 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.

Exact expanded PA statement

forall n h. n = h + h -> ~(n = 1)

Structural proof guide

Generated structural guide

A doubled predecessor cannot equal one.

Use the direct prerequisites even_odd_exclusive_pointwise, mul_comm, zero_add as previously established PA formulas.

The proof proceeds by intermediate claims (2), equality transport (1), certified simplification (1), closed numeral normalization (1).

Referenced ingredients

PA0055 even_odd_exclusive_pointwise PA000H mul_comm PA0001 zero_add

Proof neighborhood

Direct dependencies

PA0055 even_odd_exclusive_pointwise PA000H mul_comm PA0001 zero_add

Direct dependents

PA00BP odd_prime_one_not_mod_predecessor

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Alpha-v25 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.

Read the argument

Proof checkpoints

21 script commands · 3 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)
even odd exclusive pointwisemul comm
01Fix variables and assumptionsL1–4

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

  1. L1
    intro n
  2. L2
    intro h
  3. L3
    intro heven
  4. L4
    intro hone
02Establish heven_twoL5–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul comm.

  1. L5
    have heven_two : n = 2 * h
  2. L6
    trans h + h
  3. L7
    exact heven
  4. L8
    trans h * 2
  5. L9
    simp [zero_add]
  6. L10
    specialize mul_comm h
  7. L11
    specialize mul_comm 2
  8. L12
    apply mul_comm
03Establish hone_oddL13–21

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply even odd exclusive pointwise.

  1. L13
    have hone_odd : n = 2 * 0 + 1
  2. L14
    rewrite hone
  3. L15
    norm_num
  4. L16
    specialize even_odd_exclusive_pointwise n
  5. L17
    specialize even_odd_exclusive_pointwise h
  6. L18
    specialize even_odd_exclusive_pointwise 0
  7. L19
    apply even_odd_exclusive_pointwise
  8. L20
    exact heven_two
  9. L21
    exact hone_odd

Library-wide reading audit

Original exact command ledger · 21 lines
  1. 0001intro n
  2. 0002intro h
  3. 0003intro heven
  4. 0004intro hone
  5. 0005have heven_two : n = 2 * h
  6. 0006trans h + h
  7. 0007exact heven
  8. 0008trans h * 2
  9. 0009simp [zero_add]
  10. 0010specialize mul_comm h
  11. 0011specialize mul_comm 2
  12. 0012apply mul_comm
  13. 0013have hone_odd : n = 2 * 0 + 1
  14. 0014rewrite hone
  15. 0015norm_num
  16. 0016specialize even_odd_exclusive_pointwise n
  17. 0017specialize even_odd_exclusive_pointwise h
  18. 0018specialize even_odd_exclusive_pointwise 0
  19. 0019apply even_odd_exclusive_pointwise
  20. 0020exact heven_two
  21. 0021exact hone_odd