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 a. exists u v. a + m * u = a + m * vStructural proof guide
Generated structural guide
Balanced natural congruence is reflexive.
This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.
The proof proceeds by direct introduction and elimination.
Referenced ingredients
Proof neighborhood
Direct dependencies
Direct dependents
PA0026 binary_crt PA003Q coprime_mod_inverse PA005E pow_predecessor_parity_mod PA005I pow_mod_congruent PA007Q beta_product_pointwise_scale_mod PA00A0 beta_adjacent_target_pairs_product_power PA00AO inverse_prefix_zero_fixed PA00B9 beta_adjacent_unit_pairs_product_one PA00BJ prime_factorial_wilson_congruence PA00CM signed_remainder_sum_mod_twoFormal native tactic body
Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.
Read the argument
Proof checkpoints
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
02Construct an explicit witnessL3–4
03Calculate and transport equalitiesL5–5
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L5
refl