PA00BO · theorem

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.

Statement with defined notation

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

none

0 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall n h. n = h + h -> ~(n = 1)

Proof neighborhood

Direct theorem prerequisites

PA0055 even_odd_exclusive_pointwise PA000H mul_comm PA0001 zero_add

Direct theorem dependents

PA00BP odd_prime_one_not_mod_predecessor

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

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.

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)
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 defined 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