CD0006

finite_add_le_add

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The two genuine finite-order witnesses add componentwise.

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 first-order arithmetic statement

forall a b c d. (exists fms_gap_le. fms_gap_le + (a) = (b)) -> (exists fms_gap_le. fms_gap_le + (c) = (d)) -> (exists fms_gap_le. fms_gap_le + (a+c) = (b+d))

Constructive proof overview

Generated structural guide

The two genuine finite-order witnesses add componentwise.

The unchanged tactic script uses 3 declared prerequisites and contains 20 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_le_add_right Stable theorem; checked-use authorized add_le_add_left Stable theorem; checked-use authorized le_trans Stable theorem; checked-use authorized

Direct dependents

CD0007 finite_add_lt_of_lt_of_le CD0008 finite_add_lt_of_le_of_lt

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

20 script commands · 3 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 assumptionsL1–6

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro hab
  6. L6
    intro hcd
02Use earlier factsL7–16

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

  1. L7
    specialize le_trans a+c
  2. L8
    specialize le_trans b+c
  3. L9
    specialize le_trans b+d
  4. L10
    apply le_trans
  5. L11
    specialize add_le_add_right a
  6. L12
    specialize add_le_add_right b
  7. L13
    specialize add_le_add_right c
  8. L14
    apply add_le_add_right
  9. L15
    exact hab
  10. L16
    specialize add_le_add_left c
03Use earlier factsL17–20

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

  1. L17
    specialize add_le_add_left d
  2. L18
    specialize add_le_add_left b
  3. L19
    apply add_le_add_left
  4. L20
    exact hcd

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro hab
  6. 0006intro hcd
  7. 0007specialize le_trans a+c
  8. 0008specialize le_trans b+c
  9. 0009specialize le_trans b+d
  10. 0010apply le_trans
  11. 0011specialize add_le_add_right a
  12. 0012specialize add_le_add_right b
  13. 0013specialize add_le_add_right c
  14. 0014apply add_le_add_right
  15. 0015exact hab
  16. 0016specialize add_le_add_left c
  17. 0017specialize add_le_add_left d
  18. 0018specialize add_le_add_left b
  19. 0019apply add_le_add_left
  20. 0020exact hcd