PA000V

le_of_succ_le_succ

Stable checked-use theorem · independently closed

Successor order reflects to the underlying naturals.

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.

  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