PA003B

le_or_lt

Stable checked-use theorem · independently closed

Any two naturals satisfy weak order in one direction or strict order in the other.

Exact expanded PA statement

forall a b. (exists k. k + a = b) \/ exists k. k + S b = a

Structural proof guide

Generated structural guide

Any two naturals satisfy weak order in one direction or strict order in the other.

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

The proof proceeds by structural induction (2), case analysis (3), equality transport (2).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

PA0070 gauss_pointwise_signed_half_representative PA00DR odd_half_division_quotient_bounded PA00DS nonzero_remainder_division_positive_multiple_threshold PA00DT eisenstein_row_indicator_prefix_to_initial_segment

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. 0001induction a
  2. 0002intro b
  3. 0003left
  4. 0004exists b
  5. 0005apply PA3
  6. 0006induction b
  7. 0007right
  8. 0008exists a
  9. 0009trans S (a + 0)
  10. 0010apply PA4
  11. 0011congr
  12. 0012apply PA3
  13. 0013specialize IH b
  14. 0014cases IH
  15. 0015left
  16. 0016cases IH_left
  17. 0017exists x
  18. 0018rewrite PA4
  19. 0019congr
  20. 0020exact IH_left_witness
  21. 0021right
  22. 0022cases IH_right
  23. 0023exists x
  24. 0024rewrite PA4
  25. 0025congr
  26. 0026exact IH_right_witness