PA0091

euler_pair_iteration_step_short

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

Expose a strict-prefix witness whenever at least one pair remains.

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 m k. S (k + k) + S (m + m) = (S m + S m) + (k + k)

Structural proof guide

Generated structural guide

Expose a strict-prefix witness whenever at least one pair remains.

Use the direct prerequisites add_succ_left, add_comm as previously established PA formulas.

The proof proceeds by certified simplification (1).

Referenced ingredients

PA000E add_succ_left PA000F add_comm

Proof neighborhood

Direct dependencies

PA000E add_succ_left PA000F add_comm

Direct dependents

PA009V scaled_inverse_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

3 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 assumptionsL1–2

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

  1. L1
    intro m
  2. L2
    intro k
02Calculate and transport equalitiesL3–3

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

  1. L3
    simp [add_succ_left, add_comm]

Library-wide reading audit

Original exact command ledger · 3 lines
  1. 0001intro m
  2. 0002intro k
  3. 0003simp [add_succ_left, add_comm]