PA000V

le_of_succ_le_succ

Stable checked-use theorem · independently closed

Successor order reflects to the underlying naturals.

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 a b. (exists k. k + S a = S b) -> exists r. r + a = b

Structural proof guide

Generated structural guide

Successor order reflects to the underlying naturals.

This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.

The proof proceeds by case analysis (1).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

PA001K gcd_balanced_bezout_exists_up_to PA001U beta_exclusive_accumulated_product_step PA002G beta_exclusive_recode_congruence_step PA002Y beta_range_succ_extend PA0035 gcd_exists_up_to PA003D finite_lt_succ_eq_or_lt PA003G beta_prefix_sum_trace_exists PA003W beta_prefix_product_trace_exists PA0044 beta_repeat_succ_extend PA008B prime_mul_index_map_exists_up_to PA009A scaled_inverse_prefix_mate_predecessor PA00A4 prime_inverse_index_exists PA00B4 finite_bounded_nonendpoint_injective_coverage PA00BB pair_order_terminal_state_magnitude_range PA00DX eisenstein_initial_segment_bit_count_functional

Formal 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

10 script commands · 6 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 a
  2. L2
    intro b
  3. L3
    intro h
02Separate the logical casesL4–4

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L4
    cases h
03Construct an explicit witnessL5–5

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

  1. L5
    exists x
04Use earlier factsL6–6

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L6
    apply PA2
05Calculate and transport equalitiesL7–8

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

  1. L7
    trans x + S a
  2. L8
    symm
06Use earlier factsL9–10

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L9
    apply PA4
  2. L10
    exact h_witness

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004cases h
  5. 0005exists x
  6. 0006apply PA2
  7. 0007trans x + S a
  8. 0008symm
  9. 0009apply PA4
  10. 0010exact h_witness