GF0022

gaussian_ring_pair_sum_cancel

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

Cancel a common represented integer summand using ordinary natural addition 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 a b c d e f. (a+c)+(b+f)=(a+e)+(b+d) -> c+f=e+d

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 4 declared prerequisites and contains 16 exact native proof lines.

Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_left_cancel 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

GF0023 gaussian_ring_raw_add_cancel_left

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 · 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.

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 exact 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]