GF0022

gaussian_ring_pair_sum_cancel

Cancel a common represented integer summand using ordinary natural addition cancellation.

Alpha v34 checked-use · first admitted v30 · 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.

Inputs are genuine canonical signed-pair codes, not arbitrary naturals. Products start at the actual Gaussian identity, whose code is six. The factor list uses the proved prime-divisor property; irreducibility alone is not silently renamed primality. Uniqueness supplies equal lengths, a bounded bijection, and an actual unit at each match, including repeated factors. Units have empty factorizations and zero is excluded. Sorted primary representatives, Gaussian prime classification, and Eisenstein factorization are separate targets.

Exact theorem in conservative defined notation

∀ a. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. a + c + (b + f) = a + e + (b + d) → c + f = e + d

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

Definition DAG

none

Actual proof prerequisites

add_left_cancel · 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 a b c d e f. (a+c)+(b+f)=(a+e)+(b+d) -> c+f=e+d

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 · 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 assumptionsL1–7

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro e
  6. L6
    intro f
  7. L7
    intro h
02Use earlier factsL8–11

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

  1. L8
    specialize add_left_cancel (a+b)
  2. L9
    specialize add_left_cancel (c+f)
  3. L10
    specialize add_left_cancel (e+d)
  4. L11
    apply add_left_cancel
03Calculate and transport equalitiesL12–14

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

  1. L12
    trans (a+c)+(b+f)
  2. L13
    simp [add_assoc, add_comm, four_square_add_swap_right_tail]
  3. L14
    trans (a+e)+(b+d)
04Use earlier factsL15–15

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

  1. L15
    exact h
05Calculate and transport equalitiesL16–16

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

  1. L16
    simp [add_assoc, add_comm, four_square_add_swap_right_tail]

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro h
  8. 0008specialize add_left_cancel (a+b)
  9. 0009specialize add_left_cancel (c+f)
  10. 0010specialize add_left_cancel (e+d)
  11. 0011apply add_left_cancel
  12. 0012trans (a+c)+(b+f)
  13. 0013simp [add_assoc, add_comm, four_square_add_swap_right_tail]
  14. 0014trans (a+e)+(b+d)
  15. 0015exact h
  16. 0016simp [add_assoc, add_comm, four_square_add_swap_right_tail]