PA006O

lt_trans

Stable checked-use theorem · independently closed

Strict order is transitive.

Exact expanded PA statement

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

Structural proof guide

Generated structural guide

Strict order is transitive.

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

The proof proceeds by case analysis (2).

Referenced ingredients

PA0009 add_assoc PA000E add_succ_left

Proof neighborhood

Direct dependencies

PA0009 add_assoc PA000E add_succ_left

Direct dependents

PA00DS nonzero_remainder_division_positive_multiple_threshold

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 c
  4. 0004intro hab
  5. 0005intro hbc
  6. 0006cases hab
  7. 0007cases hbc
  8. 0008exists x1 + S x
  9. 0009trans x1 + (S x + S a)
  10. 0010apply add_assoc
  11. 0011trans x1 + S (x + S a)
  12. 0012congr
  13. 0013refl
  14. 0014apply add_succ_left
  15. 0015trans x1 + S b
  16. 0016congr
  17. 0017refl
  18. 0018congr
  19. 0019exact hab_witness
  20. 0020exact hbc_witness