BA0013

cf_approximation_signed_coordinates_pn

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

Cancelling the common nonnegative coordinate representatives yields the actual pn signed linear combination.

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 x y u U p n q m c d. ((x) + ((n) * (u) + (m) * (U)) = (y) + ((p) * (u) + (q) * (U))) -> p = n + c -> m = q + d -> (x) + (d) * (U) = (y) + (c) * (u)

Constructive proof overview

Generated structural guide

Cancelling the common nonnegative coordinate representatives yields the actual pn signed linear combination.

The unchanged tactic script uses 5 declared prerequisites and contains 24 exact native proof lines.

Alpha v34 checked-use · first admitted v29 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_right_cancel Stable theorem; checked-use authorized add_mul Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized four_square_add_swap_right_tail Alpha theorem; checked-use authorized

Direct dependents

BA0018 cf_approximation_signed_basis_normalization

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

24 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 assumptionsL1–10

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro u
  4. L4
    intro U
  5. L5
    intro p
  6. L6
    intro n
  7. L7
    intro q
  8. L8
    intro m
  9. L9
    intro c
  10. L10
    intro d
02Fix variables and assumptionsL11–13

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

  1. L11
    intro h
  2. L12
    intro hp
  3. L13
    intro hq
03Use earlier factsL14–17

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

  1. L14
    specialize add_right_cancel ((x) + (d) * (U))
  2. L15
    specialize add_right_cancel ((y) + (c) * (u))
  3. L16
    specialize add_right_cancel (n * u + q * U)
  4. L17
    apply add_right_cancel
04Calculate and transport equalitiesL18–21

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

  1. L18
    trans x + (n * u + m * U)
  2. L19
    rewrite hq
  3. L20
    simp [add_mul, add_assoc, add_comm, four_square_add_swap_right_tail]
  4. L21
    trans y + (p * u + q * U)
05Use earlier factsL22–22

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

  1. L22
    exact h
06Calculate and transport equalitiesL23–24

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

  1. L23
    rewrite hp
  2. L24
    simp [add_mul, add_assoc, add_comm, four_square_add_swap_right_tail]

Library-wide reading audit

Original exact command ledger · 24 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro u
  4. 0004intro U
  5. 0005intro p
  6. 0006intro n
  7. 0007intro q
  8. 0008intro m
  9. 0009intro c
  10. 0010intro d
  11. 0011intro h
  12. 0012intro hp
  13. 0013intro hq
  14. 0014specialize add_right_cancel ((x) + (d) * (U))
  15. 0015specialize add_right_cancel ((y) + (c) * (u))
  16. 0016specialize add_right_cancel (n * u + q * U)
  17. 0017apply add_right_cancel
  18. 0018trans x + (n * u + m * U)
  19. 0019rewrite hq
  20. 0020simp [add_mul, add_assoc, add_comm, four_square_add_swap_right_tail]
  21. 0021trans y + (p * u + q * U)
  22. 0022exact h
  23. 0023rewrite hp
  24. 0024simp [add_mul, add_assoc, add_comm, four_square_add_swap_right_tail]