SL0002

odd_predecessor_double_half

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

An odd successor has predecessor equal to twice its odd half.

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 first-order arithmetic statement

forall p n h. p = S n -> p = 2 * h + 1 -> n = h + h

Constructive proof overview

Generated structural guide

An odd successor has predecessor equal to twice its odd half.

The unchanged tactic script uses 2 declared prerequisites and contains 25 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

mul_comm Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized

Direct dependents

SL0004 quadratic_supplement_minus_one_residue_iff_mod_four_one SL0005 quadratic_supplement_minus_one_nonresidue_iff_mod_four_three

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

25 script commands · 4 reading checkpoints · 3 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro h
  4. L4
    intro hpredecessor
  5. L5
    intro hodd
02Establish hdoubleL6–11

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

  1. L6
    have hdouble : h + h = 2 * h
  2. L7
    trans h * 2
  3. L8
    simp [zero_add]
  4. L9
    specialize mul_comm h
  5. L10
    specialize mul_comm 2
  6. L11
    apply mul_comm
03Establish hsuccessorsL12–18

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

  1. L12
    have hsuccessors : S n = S (2 * h)
  2. L13
    trans p
  3. L14
    symm
  4. L15
    exact hpredecessor
  5. L16
    trans 2 * h + 1
  6. L17
    exact hodd
  7. L18
    simp
04Establish hpredecessorsL19–25

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply PA2.

  1. L19
    have hpredecessors : n = 2 * h
  2. L20
    apply PA2
  3. L21
    exact hsuccessors
  4. L22
    trans 2 * h
  5. L23
    exact hpredecessors
  6. L24
    symm
  7. L25
    exact hdouble

Library-wide reading audit

Original exact command ledger · 25 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro h
  4. 0004intro hpredecessor
  5. 0005intro hodd
  6. 0006have hdouble : h + h = 2 * h
  7. 0007trans h * 2
  8. 0008simp [zero_add]
  9. 0009specialize mul_comm h
  10. 0010specialize mul_comm 2
  11. 0011apply mul_comm
  12. 0012have hsuccessors : S n = S (2 * h)
  13. 0013trans p
  14. 0014symm
  15. 0015exact hpredecessor
  16. 0016trans 2 * h + 1
  17. 0017exact hodd
  18. 0018simp
  19. 0019have hpredecessors : n = 2 * h
  20. 0020apply PA2
  21. 0021exact hsuccessors
  22. 0022trans 2 * h
  23. 0023exact hpredecessors
  24. 0024symm
  25. 0025exact hdouble