SL0002 · theorem body

odd_predecessor_double_half

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

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

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

none

In local proof propositions

none
Exact expanded first-order statement
forall p n h. p = S n -> p = 2 * h + 1 -> n = h + h

Proof neighborhood

Direct theorem prerequisites

mul_comm · Stable closed zero_add · Stable closed

Direct theorem dependents

SL0004 quadratic_supplement_minus_one_residue_iff_mod_four_one SL0005 quadratic_supplement_minus_one_nonresidue_iff_mod_four_three

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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