GF0023

gaussian_ring_raw_add_cancel_left

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

A common Gaussian summand cancels in both actual represented integer coordinates.

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 ap an bp bn cp cn dp dn ep en fp fn. (((((((ap) + (cp))) + (((an) + (en)))) = ((((ap) + (ep))) + (((an) + (cn))))) /\ (((((bp) + (dp))) + (((bn) + (fn)))) = ((((bp) + (fp))) + (((bn) + (dn))))))) -> (((((cp) + (en)) = ((ep) + (cn))) /\ (((dp) + (fn)) = ((fp) + (dn)))))

Constructive proof overview

Generated structural guide

A common Gaussian summand cancels in both actual represented integer coordinates.

The unchanged tactic script uses 1 declared prerequisite and contains 31 exact native proof lines.

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

Proof neighborhood

Direct dependencies

GF0022 gaussian_ring_pair_sum_cancel

Direct dependents

GF002D gaussian_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

31 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.

Named ingredients (1)
gaussian ring pair sum cancel
01Fix variables and assumptionsL1–10

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

  1. L1
    intro ap
  2. L2
    intro an
  3. L3
    intro bp
  4. L4
    intro bn
  5. L5
    intro cp
  6. L6
    intro cn
  7. L7
    intro dp
  8. L8
    intro dn
  9. L9
    intro ep
  10. L10
    intro en
02Fix variables and assumptionsL11–13

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

  1. L11
    intro fp
  2. L12
    intro fn
  3. L13
    intro h
03Separate the logical casesL14–15

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L14
    cases h
  2. L15
    split
04Use earlier factsL16–25

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

  1. L16
    specialize gaussian_ring_pair_sum_cancel (ap)
  2. L17
    specialize gaussian_ring_pair_sum_cancel (an)
  3. L18
    specialize gaussian_ring_pair_sum_cancel (cp)
  4. L19
    specialize gaussian_ring_pair_sum_cancel (cn)
  5. L20
    specialize gaussian_ring_pair_sum_cancel (ep)
  6. L21
    specialize gaussian_ring_pair_sum_cancel (en)
  7. L22
    apply gaussian_ring_pair_sum_cancel
  8. L23
    exact h_left
  9. L24
    specialize gaussian_ring_pair_sum_cancel (bp)
  10. L25
    specialize gaussian_ring_pair_sum_cancel (bn)
05Use earlier factsL26–31

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

  1. L26
    specialize gaussian_ring_pair_sum_cancel (dp)
  2. L27
    specialize gaussian_ring_pair_sum_cancel (dn)
  3. L28
    specialize gaussian_ring_pair_sum_cancel (fp)
  4. L29
    specialize gaussian_ring_pair_sum_cancel (fn)
  5. L30
    apply gaussian_ring_pair_sum_cancel
  6. L31
    exact h_right

Library-wide reading audit

Original exact command ledger · 31 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro cp
  6. 0006intro cn
  7. 0007intro dp
  8. 0008intro dn
  9. 0009intro ep
  10. 0010intro en
  11. 0011intro fp
  12. 0012intro fn
  13. 0013intro h
  14. 0014cases h
  15. 0015split
  16. 0016specialize gaussian_ring_pair_sum_cancel (ap)
  17. 0017specialize gaussian_ring_pair_sum_cancel (an)
  18. 0018specialize gaussian_ring_pair_sum_cancel (cp)
  19. 0019specialize gaussian_ring_pair_sum_cancel (cn)
  20. 0020specialize gaussian_ring_pair_sum_cancel (ep)
  21. 0021specialize gaussian_ring_pair_sum_cancel (en)
  22. 0022apply gaussian_ring_pair_sum_cancel
  23. 0023exact h_left
  24. 0024specialize gaussian_ring_pair_sum_cancel (bp)
  25. 0025specialize gaussian_ring_pair_sum_cancel (bn)
  26. 0026specialize gaussian_ring_pair_sum_cancel (dp)
  27. 0027specialize gaussian_ring_pair_sum_cancel (dn)
  28. 0028specialize gaussian_ring_pair_sum_cancel (fp)
  29. 0029specialize gaussian_ring_pair_sum_cancel (fn)
  30. 0030apply gaussian_ring_pair_sum_cancel
  31. 0031exact h_right