PA00C2

odd_half_strictly_below_modulus

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

The half of an odd modulus p=2h+1 is strictly below p.

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 p h. p = 2 * h + 1 -> (exists frp_gap_magnitude_half_below. frp_gap_magnitude_half_below + S h = p)

Structural proof guide

Generated structural guide

The half of an odd modulus p=2h+1 is strictly below p.

Use the direct prerequisites mul_succ_left, mul_zero_left, zero_add, add_succ_left, add_assoc as previously established PA formulas.

The proof proceeds by equality transport (1), certified simplification (1).

Referenced ingredients

PA000G mul_succ_left PA000D mul_zero_left PA0001 zero_add PA000E add_succ_left PA0009 add_assoc

Proof neighborhood

Direct dependencies

PA000G mul_succ_left PA000D mul_zero_left PA0001 zero_add PA000E add_succ_left PA0009 add_assoc

Direct dependents

PA00C6 odd_signed_division_branch_exact PA00D8 distinct_odd_prime_half_products_ne PA00DP distinct_primes_own_odd_half_scaled_remainder_nonzero

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

6 script commands · 3 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro hp
02Construct an explicit witnessL4–4

Supply the displayed value, then prove that it has the required property.

  1. L4
    exists h
03Calculate and transport equalitiesL5–6

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

  1. L5
    rewrite hp
  2. L6
    simp [mul_succ_left, mul_zero_left, zero_add, add_succ_left, add_assoc]

Library-wide reading audit

Original exact command ledger · 6 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro hp
  4. 0004exists h
  5. 0005rewrite hp
  6. 0006simp [mul_succ_left, mul_zero_left, zero_add, add_succ_left, add_assoc]