DL009B

matrix_integer_nonzero_pair_transport

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

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

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 p n P N. p + N = P + n -> ~(p = n) -> ~(P = N)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 16 exact native proof lines.

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

Proof neighborhood

Direct dependencies

add_right_cancel Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

DL009C matrix_integer_nonzero_minor_transport

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

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.

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 exact 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