DL009B

matrix_integer_nonzero_pair_transport

Nonzeroness of the represented integer is preserved by cross-sum equality, using actual additive cancellation.

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. ∀ P. ∀ N. p + N = P + n → ¬p = n → ¬P = N

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

Definition DAG

none

Actual proof prerequisites

add_right_cancel · checked external prerequisiteadd_comm · checked external prerequisite
Original expanded first-order statement 
forall p n P N. p + N = P + n -> ~(p = n) -> ~(P = N)

Complete tactic proof in conservative notation

All 16 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

16 script commands · 6 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 assumptionsL1–7

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro P
  4. L4
    intro N
  5. L5
    intro hbalance
  6. L6
    intro hnonzero
  7. L7
    intro hzero
02Use earlier factsL8–12

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

  1. L8
    apply hnonzero
  2. L9
    specialize add_right_cancel (p)
  3. L10
    specialize add_right_cancel (n)
  4. L11
    specialize add_right_cancel (N)
  5. L12
    apply add_right_cancel
03Calculate and transport equalitiesL13–13

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

  1. L13
    trans P + n
04Use earlier factsL14–14

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

  1. L14
    exact hbalance
05Calculate and transport equalitiesL15–15

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

  1. L15
    rewrite hzero
06Use earlier factsL16–16

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

  1. L16
    apply add_comm

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro P
  4. 0004intro N
  5. 0005intro hbalance
  6. 0006intro hnonzero
  7. 0007intro hzero
  8. 0008apply hnonzero
  9. 0009specialize add_right_cancel (p)
  10. 0010specialize add_right_cancel (n)
  11. 0011specialize add_right_cancel (N)
  12. 0012apply add_right_cancel
  13. 0013trans P + n
  14. 0014exact hbalance
  15. 0015rewrite hzero
  16. 0016apply add_comm