DL006D

integer_span_pair_add_congruence

Component addition represents the sum of integer pairs even when either input has a different noncanonical positive/negative encoding.

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.

This branch proves the finite determinant/rank/span substrate. It does not claim Smith or Hermite normal form, lattice index equals determinant, determinant multiplicativity, lattice reduction, or geometry-of-numbers theorems.

Exact theorem in conservative defined notation

∀ p. ∀ n. ∀ q. ∀ m. ∀ a. ∀ b. ∀ c. ∀ d. p + b = a + n → q + d = c + m → p + q + (b + d) = a + c + (n + m)

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

Definition DAG

none

Actual proof prerequisites

add_shuffle_middle · checked external prerequisite
Original expanded first-order statement 
forall p n q m a b c d. p + b = a + n -> q + d = c + m -> (p + q) + (b + d) = (a + c) + (n + m)

Complete tactic proof in conservative notation

All 17 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

17 script commands · 5 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 assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro q
  4. L4
    intro m
  5. L5
    intro a
  6. L6
    intro b
  7. L7
    intro c
  8. L8
    intro d
  9. L9
    intro hfirst
  10. L10
    intro hsecond
02Calculate and transport equalitiesL11–11

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L11
    trans (p + b) + (q + d)
03Use earlier factsL12–12

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

  1. L12
    apply add_shuffle_middle
04Calculate and transport equalitiesL13–14

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L13
    trans (a + n) + (c + m)
  2. L14
    congr
05Use earlier factsL15–17

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

  1. L15
    exact hfirst
  2. L16
    exact hsecond
  3. L17
    apply add_shuffle_middle

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro q
  4. 0004intro m
  5. 0005intro a
  6. 0006intro b
  7. 0007intro c
  8. 0008intro d
  9. 0009intro hfirst
  10. 0010intro hsecond
  11. 0011trans (p + b) + (q + d)
  12. 0012apply add_shuffle_middle
  13. 0013trans (a + n) + (c + m)
  14. 0014congr
  15. 0015exact hfirst
  16. 0016exact hsecond
  17. 0017apply add_shuffle_middle