BA0015

cf_approximation_signed_coordinates_nn

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

Alpha v34 checked-use · first admitted v29 · 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.

The initial 0/1 convergent is included: u is natural, not necessarily positive. Comparison denominators are strictly smaller and positive. Signed competitors are represented by an arbitrary difference rp−rn. Approximation inequalities are proved from the trace, never stored as assumptions in Convergent.

Exact theorem in conservative defined notation

∀ x. ∀ y. ∀ u. ∀ U. ∀ p. ∀ n. ∀ q. ∀ m. ∀ c. ∀ d. x + (n · u + m · U) = y + (p · u + q · U) → n = p + c → m = q + d → x + (c · u + d · U) = y

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

Definition DAG

none

Actual proof prerequisites

add_right_cancel · checked external prerequisiteadd_mul · checked external prerequisiteadd_assoc · checked external prerequisiteadd_comm · checked external prerequisitefour_square_add_swap_right_tail · checked external prerequisite
Original expanded first-order statement 
forall x y u U p n q m c d. ((x) + ((n) * (u) + (m) * (U)) = (y) + ((p) * (u) + (q) * (U))) -> n = p + c -> m = q + d -> (x) + ((c) * (u) + (d) * (U)) = (y)

Complete tactic proof in conservative notation

All 24 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

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.

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 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) + ((c) * (u) + (d) * (U)))
  2. L15
    specialize add_right_cancel ((y))
  3. L16
    specialize add_right_cancel (p * u + q * U)
  4. L17
    apply add_right_cancel
04Calculate and transport equalitiesL18–22

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 hp
  3. L20
    rewrite hq
  4. L21
    simp [add_mul, add_assoc, add_comm, four_square_add_swap_right_tail]
  5. L22
    trans y + (p * u + q * U)
05Use earlier factsL23–23

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

  1. L23
    exact h
06Calculate and transport equalitiesL24–24

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

  1. L24
    simp [add_mul, add_assoc, add_comm, four_square_add_swap_right_tail]

Library-wide reading audit

Original defined 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) + ((c) * (u) + (d) * (U)))
  15. 0015specialize add_right_cancel ((y))
  16. 0016specialize add_right_cancel (p * u + q * U)
  17. 0017apply add_right_cancel
  18. 0018trans x + (n * u + m * U)
  19. 0019rewrite hp
  20. 0020rewrite hq
  21. 0021simp [add_mul, add_assoc, add_comm, four_square_add_swap_right_tail]
  22. 0022trans y + (p * u + q * U)
  23. 0023exact h
  24. 0024simp [add_mul, add_assoc, add_comm, four_square_add_swap_right_tail]