CD0006

finite_add_le_add

The two genuine finite-order witnesses add componentwise.

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

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.

Sets are complete characteristic-bit codes with actual finite cardinality witnesses. The proof constructs translations and the sumset; no finite-choice oracle, supplied cardinality conclusion, or unproved polynomial-method premise is used.

Exact theorem in conservative defined notation

∀ a. ∀ b. ∀ c. ∀ d. Le(a,b) → Le(c,d) → Le(a + c,b + d)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Le(a,b) · 3

Actual proof prerequisites

add_le_add_right · checked external prerequisiteadd_le_add_left · checked external prerequisitele_trans · checked external prerequisite
Original expanded first-order 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))

Complete tactic proof in conservative notation

All 20 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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