PA009U

pair_order_double_succ_length

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

Normalize the two-entry successor length into the next doubled pair count.

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 l m. l = m + m -> S (S l) = S m + S m

Structural proof guide

Generated structural guide

Normalize the two-entry successor length into the next doubled pair count.

Use the direct prerequisites add_succ_left as previously established PA formulas.

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

Referenced ingredients

PA000E add_succ_left

Proof neighborhood

Direct dependencies

PA000E add_succ_left

Direct dependents

PA009V scaled_inverse_pair_order_paired_iteration PA00B1 prime_pair_order_paired_iteration

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 · 2 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 l
  2. L2
    intro m
  3. L3
    intro hl
02Calculate and transport equalitiesL4–6

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

  1. L4
    rewrite hl
  2. L5
    symm
  3. L6
    simp [add_succ_left]

Library-wide reading audit

Original exact command ledger · 6 lines
  1. 0001intro l
  2. 0002intro m
  3. 0003intro hl
  4. 0004rewrite hl
  5. 0005symm
  6. 0006simp [add_succ_left]