CD0007

finite_add_lt_of_lt_of_le

A strict left inequality and weak right inequality give a strict sum inequality.

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. Lt(a,b) → Le(c,d) → Lt(a + c,b + d)

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

Definition DAG

Le(a,b) · 2Lt(a,b) · 2

Actual proof prerequisites

finite_add_le_addadd_succ_left · checked external prerequisite
Original expanded first-order statement 
forall a b c d. (exists fms_gap_lt. fms_gap_lt + S (a) = (b)) -> (exists fms_gap_le. fms_gap_le + (c) = (d)) -> (exists fms_gap_lt. fms_gap_lt + S (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 · 2 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.

Named ingredients (1)
finite add le add
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
02Establish htL7–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply finite add le add.

  1. L7
  2. L8
    specialize finite_add_le_add S a
  3. L9
    specialize finite_add_le_add b
  4. L10
    specialize finite_add_le_add c
  5. L11
    specialize finite_add_le_add d
  6. L12
    apply finite_add_le_add
  7. L13
    exact hab
  8. L14
    exact hcd
03Establish heL15–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add succ left.

  1. L15
    have he : (S a)+c=S(a+c)
  2. L16
    specialize add_succ_left a
  3. L17
    specialize add_succ_left c
  4. L18
    apply add_succ_left
  5. L19
    rewrite he at ht
  6. L20
    exact ht

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. 0007have ht : Le(S a + c,b + d)
  8. 0008specialize finite_add_le_add S a
  9. 0009specialize finite_add_le_add b
  10. 0010specialize finite_add_le_add c
  11. 0011specialize finite_add_le_add d
  12. 0012apply finite_add_le_add
  13. 0013exact hab
  14. 0014exact hcd
  15. 0015have he : (S a)+c=S(a+c)
  16. 0016specialize add_succ_left a
  17. 0017specialize add_succ_left c
  18. 0018apply add_succ_left
  19. 0019rewrite he at ht
  20. 0020exact ht