GF003C

gaussian_multiply_cancel_right

A nonzero common right Gaussian factor cancels in the actual canonical multiplication graph.

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. ∀ t. ¬c = 0 → GMul(a,c,t) → GMul(b,c,t) → a = b

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

Definition DAG

GMul(a,b,c) · 2

Actual proof prerequisites

gaussian_multiply_cancel_leftgaussian_multiply_commutative
Original expanded first-order statement 
forall a b c t. ~(c=0) -> (exists ge_first_rp_cancel_right_product_first ge_first_rn_cancel_right_product_first ge_first_ip_cancel_right_product_first ge_first_in_cancel_right_product_first ge_second_rp_cancel_right_product_first ge_second_rn_cancel_right_product_first ge_second_ip_cancel_right_product_first ge_second_in_cancel_right_product_first. ((exists ge_representation_real_code_cancel_right_product_firstfirst ge_representation_imaginary_code_cancel_right_product_firstfirst. (((a) = ((ge_representation_real_code_cancel_right_product_firstfirst) + (ge_representation_imaginary_code_cancel_right_product_firstfirst)) * S ((ge_representation_real_code_cancel_right_product_firstfirst) + (ge_representation_imaginary_code_cancel_right_product_firstfirst)) + ((ge_representation_imaginary_code_cancel_right_product_firstfirst) + (ge_representation_imaginary_code_cancel_right_product_firstfirst))) /\ ((exists ge_balance_positive_cancel_right_product_firstfirstreal ge_balance_negative_cancel_right_product_firstfirstreal. (((((ge_representation_real_code_cancel_right_product_firstfirst) = 2 * (ge_balance_positive_cancel_right_product_firstfirstreal) /\ (ge_balance_negative_cancel_right_product_firstfirstreal) = 0) \/ exists ge_signed_half_cancel_right_product_firstfirstrealdecode. (((ge_representation_real_code_cancel_right_product_firstfirst) = 2 * ge_signed_half_cancel_right_product_firstfirstrealdecode + 1 /\ (ge_balance_positive_cancel_right_product_firstfirstreal) = 0) /\ (ge_balance_negative_cancel_right_product_firstfirstreal) = S ge_signed_half_cancel_right_product_firstfirstrealdecode))) /\ ((ge_first_rp_cancel_right_product_first) + ge_balance_negative_cancel_right_product_firstfirstreal = (ge_first_rn_cancel_right_product_first) + ge_balance_positive_cancel_right_product_firstfirstreal))) /\ (exists ge_balance_positive_cancel_right_product_firstfirstimaginary ge_balance_negative_cancel_right_product_firstfirstimaginary. (((((ge_representation_imaginary_code_cancel_right_product_firstfirst) = 2 * (ge_balance_positive_cancel_right_product_firstfirstimaginary) /\ (ge_balance_negative_cancel_right_product_firstfirstimaginary) = 0) \/ exists ge_signed_half_cancel_right_product_firstfirstimaginarydecode. (((ge_representation_imaginary_code_cancel_right_product_firstfirst) = 2 * ge_signed_half_cancel_right_product_firstfirstimaginarydecode + 1 /\ (ge_balance_positive_cancel_right_product_firstfirstimaginary) = 0) /\ (ge_balance_negative_cancel_right_product_firstfirstimaginary) = S ge_signed_half_cancel_right_product_firstfirstimaginarydecode))) /\ ((ge_first_ip_cancel_right_product_first) + ge_balance_negative_cancel_right_product_firstfirstimaginary = (ge_first_in_cancel_right_product_first) + ge_balance_positive_cancel_right_product_firstfirstimaginary)))))) /\ ((exists ge_representation_real_code_cancel_right_product_firstsecond ge_representation_imaginary_code_cancel_right_product_firstsecond. (((c) = ((ge_representation_real_code_cancel_right_product_firstsecond) + (ge_representation_imaginary_code_cancel_right_product_firstsecond)) * S ((ge_representation_real_code_cancel_right_product_firstsecond) + (ge_representation_imaginary_code_cancel_right_product_firstsecond)) + ((ge_representation_imaginary_code_cancel_right_product_firstsecond) + (ge_representation_imaginary_code_cancel_right_product_firstsecond))) /\ ((exists ge_balance_positive_cancel_right_product_firstsecondreal ge_balance_negative_cancel_right_product_firstsecondreal. (((((ge_representation_real_code_cancel_right_product_firstsecond) = 2 * (ge_balance_positive_cancel_right_product_firstsecondreal) /\ (ge_balance_negative_cancel_right_product_firstsecondreal) = 0) \/ exists ge_signed_half_cancel_right_product_firstsecondrealdecode. (((ge_representation_real_code_cancel_right_product_firstsecond) = 2 * ge_signed_half_cancel_right_product_firstsecondrealdecode + 1 /\ (ge_balance_positive_cancel_right_product_firstsecondreal) = 0) /\ (ge_balance_negative_cancel_right_product_firstsecondreal) = S ge_signed_half_cancel_right_product_firstsecondrealdecode))) /\ ((ge_second_rp_cancel_right_product_first) + ge_balance_negative_cancel_right_product_firstsecondreal = (ge_second_rn_cancel_right_product_first) + ge_balance_positive_cancel_right_product_firstsecondreal))) /\ (exists ge_balance_positive_cancel_right_product_firstsecondimaginary ge_balance_negative_cancel_right_product_firstsecondimaginary. (((((ge_representation_imaginary_code_cancel_right_product_firstsecond) = 2 * (ge_balance_positive_cancel_right_product_firstsecondimaginary) /\ (ge_balance_negative_cancel_right_product_firstsecondimaginary) = 0) \/ exists ge_signed_half_cancel_right_product_firstsecondimaginarydecode. (((ge_representation_imaginary_code_cancel_right_product_firstsecond) = 2 * ge_signed_half_cancel_right_product_firstsecondimaginarydecode + 1 /\ (ge_balance_positive_cancel_right_product_firstsecondimaginary) = 0) /\ (ge_balance_negative_cancel_right_product_firstsecondimaginary) = S ge_signed_half_cancel_right_product_firstsecondimaginarydecode))) /\ ((ge_second_ip_cancel_right_product_first) + ge_balance_negative_cancel_right_product_firstsecondimaginary = (ge_second_in_cancel_right_product_first) + ge_balance_positive_cancel_right_product_firstsecondimaginary)))))) /\ (exists ge_representation_real_code_cancel_right_product_firstoutput ge_representation_imaginary_code_cancel_right_product_firstoutput. (((t) = ((ge_representation_real_code_cancel_right_product_firstoutput) + (ge_representation_imaginary_code_cancel_right_product_firstoutput)) * S ((ge_representation_real_code_cancel_right_product_firstoutput) + (ge_representation_imaginary_code_cancel_right_product_firstoutput)) + ((ge_representation_imaginary_code_cancel_right_product_firstoutput) + (ge_representation_imaginary_code_cancel_right_product_firstoutput))) /\ ((exists ge_balance_positive_cancel_right_product_firstoutputreal ge_balance_negative_cancel_right_product_firstoutputreal. (((((ge_representation_real_code_cancel_right_product_firstoutput) = 2 * (ge_balance_positive_cancel_right_product_firstoutputreal) /\ (ge_balance_negative_cancel_right_product_firstoutputreal) = 0) \/ exists ge_signed_half_cancel_right_product_firstoutputrealdecode. (((ge_representation_real_code_cancel_right_product_firstoutput) = 2 * ge_signed_half_cancel_right_product_firstoutputrealdecode + 1 /\ (ge_balance_positive_cancel_right_product_firstoutputreal) = 0) /\ (ge_balance_negative_cancel_right_product_firstoutputreal) = S ge_signed_half_cancel_right_product_firstoutputrealdecode))) /\ ((((((((ge_first_rp_cancel_right_product_first) * (ge_second_rp_cancel_right_product_first))) + (((ge_first_rn_cancel_right_product_first) * (ge_second_rn_cancel_right_product_first))))) + (((((ge_first_ip_cancel_right_product_first) * (ge_second_in_cancel_right_product_first))) + (((ge_first_in_cancel_right_product_first) * (ge_second_ip_cancel_right_product_first))))))) + ge_balance_negative_cancel_right_product_firstoutputreal = (((((((ge_first_rp_cancel_right_product_first) * (ge_second_rn_cancel_right_product_first))) + (((ge_first_rn_cancel_right_product_first) * (ge_second_rp_cancel_right_product_first))))) + (((((ge_first_ip_cancel_right_product_first) * (ge_second_ip_cancel_right_product_first))) + (((ge_first_in_cancel_right_product_first) * (ge_second_in_cancel_right_product_first))))))) + ge_balance_positive_cancel_right_product_firstoutputreal))) /\ (exists ge_balance_positive_cancel_right_product_firstoutputimaginary ge_balance_negative_cancel_right_product_firstoutputimaginary. (((((ge_representation_imaginary_code_cancel_right_product_firstoutput) = 2 * (ge_balance_positive_cancel_right_product_firstoutputimaginary) /\ (ge_balance_negative_cancel_right_product_firstoutputimaginary) = 0) \/ exists ge_signed_half_cancel_right_product_firstoutputimaginarydecode. (((ge_representation_imaginary_code_cancel_right_product_firstoutput) = 2 * ge_signed_half_cancel_right_product_firstoutputimaginarydecode + 1 /\ (ge_balance_positive_cancel_right_product_firstoutputimaginary) = 0) /\ (ge_balance_negative_cancel_right_product_firstoutputimaginary) = S ge_signed_half_cancel_right_product_firstoutputimaginarydecode))) /\ ((((((((ge_first_rp_cancel_right_product_first) * (ge_second_ip_cancel_right_product_first))) + (((ge_first_rn_cancel_right_product_first) * (ge_second_in_cancel_right_product_first))))) + (((((ge_first_ip_cancel_right_product_first) * (ge_second_rp_cancel_right_product_first))) + (((ge_first_in_cancel_right_product_first) * (ge_second_rn_cancel_right_product_first))))))) + ge_balance_negative_cancel_right_product_firstoutputimaginary = (((((((ge_first_rp_cancel_right_product_first) * (ge_second_in_cancel_right_product_first))) + (((ge_first_rn_cancel_right_product_first) * (ge_second_ip_cancel_right_product_first))))) + (((((ge_first_ip_cancel_right_product_first) * (ge_second_rn_cancel_right_product_first))) + (((ge_first_in_cancel_right_product_first) * (ge_second_rp_cancel_right_product_first))))))) + ge_balance_positive_cancel_right_product_firstoutputimaginary))))))))) -> (exists ge_first_rp_cancel_right_product_second ge_first_rn_cancel_right_product_second ge_first_ip_cancel_right_product_second ge_first_in_cancel_right_product_second ge_second_rp_cancel_right_product_second ge_second_rn_cancel_right_product_second ge_second_ip_cancel_right_product_second ge_second_in_cancel_right_product_second. ((exists ge_representation_real_code_cancel_right_product_secondfirst ge_representation_imaginary_code_cancel_right_product_secondfirst. (((b) = ((ge_representation_real_code_cancel_right_product_secondfirst) + (ge_representation_imaginary_code_cancel_right_product_secondfirst)) * S ((ge_representation_real_code_cancel_right_product_secondfirst) + (ge_representation_imaginary_code_cancel_right_product_secondfirst)) + ((ge_representation_imaginary_code_cancel_right_product_secondfirst) + (ge_representation_imaginary_code_cancel_right_product_secondfirst))) /\ ((exists ge_balance_positive_cancel_right_product_secondfirstreal ge_balance_negative_cancel_right_product_secondfirstreal. (((((ge_representation_real_code_cancel_right_product_secondfirst) = 2 * (ge_balance_positive_cancel_right_product_secondfirstreal) /\ (ge_balance_negative_cancel_right_product_secondfirstreal) = 0) \/ exists ge_signed_half_cancel_right_product_secondfirstrealdecode. (((ge_representation_real_code_cancel_right_product_secondfirst) = 2 * ge_signed_half_cancel_right_product_secondfirstrealdecode + 1 /\ (ge_balance_positive_cancel_right_product_secondfirstreal) = 0) /\ (ge_balance_negative_cancel_right_product_secondfirstreal) = S ge_signed_half_cancel_right_product_secondfirstrealdecode))) /\ ((ge_first_rp_cancel_right_product_second) + ge_balance_negative_cancel_right_product_secondfirstreal = (ge_first_rn_cancel_right_product_second) + ge_balance_positive_cancel_right_product_secondfirstreal))) /\ (exists ge_balance_positive_cancel_right_product_secondfirstimaginary ge_balance_negative_cancel_right_product_secondfirstimaginary. (((((ge_representation_imaginary_code_cancel_right_product_secondfirst) = 2 * (ge_balance_positive_cancel_right_product_secondfirstimaginary) /\ (ge_balance_negative_cancel_right_product_secondfirstimaginary) = 0) \/ exists ge_signed_half_cancel_right_product_secondfirstimaginarydecode. (((ge_representation_imaginary_code_cancel_right_product_secondfirst) = 2 * ge_signed_half_cancel_right_product_secondfirstimaginarydecode + 1 /\ (ge_balance_positive_cancel_right_product_secondfirstimaginary) = 0) /\ (ge_balance_negative_cancel_right_product_secondfirstimaginary) = S ge_signed_half_cancel_right_product_secondfirstimaginarydecode))) /\ ((ge_first_ip_cancel_right_product_second) + ge_balance_negative_cancel_right_product_secondfirstimaginary = (ge_first_in_cancel_right_product_second) + ge_balance_positive_cancel_right_product_secondfirstimaginary)))))) /\ ((exists ge_representation_real_code_cancel_right_product_secondsecond ge_representation_imaginary_code_cancel_right_product_secondsecond. (((c) = ((ge_representation_real_code_cancel_right_product_secondsecond) + (ge_representation_imaginary_code_cancel_right_product_secondsecond)) * S ((ge_representation_real_code_cancel_right_product_secondsecond) + (ge_representation_imaginary_code_cancel_right_product_secondsecond)) + ((ge_representation_imaginary_code_cancel_right_product_secondsecond) + (ge_representation_imaginary_code_cancel_right_product_secondsecond))) /\ ((exists ge_balance_positive_cancel_right_product_secondsecondreal ge_balance_negative_cancel_right_product_secondsecondreal. (((((ge_representation_real_code_cancel_right_product_secondsecond) = 2 * (ge_balance_positive_cancel_right_product_secondsecondreal) /\ (ge_balance_negative_cancel_right_product_secondsecondreal) = 0) \/ exists ge_signed_half_cancel_right_product_secondsecondrealdecode. (((ge_representation_real_code_cancel_right_product_secondsecond) = 2 * ge_signed_half_cancel_right_product_secondsecondrealdecode + 1 /\ (ge_balance_positive_cancel_right_product_secondsecondreal) = 0) /\ (ge_balance_negative_cancel_right_product_secondsecondreal) = S ge_signed_half_cancel_right_product_secondsecondrealdecode))) /\ ((ge_second_rp_cancel_right_product_second) + ge_balance_negative_cancel_right_product_secondsecondreal = (ge_second_rn_cancel_right_product_second) + ge_balance_positive_cancel_right_product_secondsecondreal))) /\ (exists ge_balance_positive_cancel_right_product_secondsecondimaginary ge_balance_negative_cancel_right_product_secondsecondimaginary. (((((ge_representation_imaginary_code_cancel_right_product_secondsecond) = 2 * (ge_balance_positive_cancel_right_product_secondsecondimaginary) /\ (ge_balance_negative_cancel_right_product_secondsecondimaginary) = 0) \/ exists ge_signed_half_cancel_right_product_secondsecondimaginarydecode. (((ge_representation_imaginary_code_cancel_right_product_secondsecond) = 2 * ge_signed_half_cancel_right_product_secondsecondimaginarydecode + 1 /\ (ge_balance_positive_cancel_right_product_secondsecondimaginary) = 0) /\ (ge_balance_negative_cancel_right_product_secondsecondimaginary) = S ge_signed_half_cancel_right_product_secondsecondimaginarydecode))) /\ ((ge_second_ip_cancel_right_product_second) + ge_balance_negative_cancel_right_product_secondsecondimaginary = (ge_second_in_cancel_right_product_second) + ge_balance_positive_cancel_right_product_secondsecondimaginary)))))) /\ (exists ge_representation_real_code_cancel_right_product_secondoutput ge_representation_imaginary_code_cancel_right_product_secondoutput. (((t) = ((ge_representation_real_code_cancel_right_product_secondoutput) + (ge_representation_imaginary_code_cancel_right_product_secondoutput)) * S ((ge_representation_real_code_cancel_right_product_secondoutput) + (ge_representation_imaginary_code_cancel_right_product_secondoutput)) + ((ge_representation_imaginary_code_cancel_right_product_secondoutput) + (ge_representation_imaginary_code_cancel_right_product_secondoutput))) /\ ((exists ge_balance_positive_cancel_right_product_secondoutputreal ge_balance_negative_cancel_right_product_secondoutputreal. (((((ge_representation_real_code_cancel_right_product_secondoutput) = 2 * (ge_balance_positive_cancel_right_product_secondoutputreal) /\ (ge_balance_negative_cancel_right_product_secondoutputreal) = 0) \/ exists ge_signed_half_cancel_right_product_secondoutputrealdecode. (((ge_representation_real_code_cancel_right_product_secondoutput) = 2 * ge_signed_half_cancel_right_product_secondoutputrealdecode + 1 /\ (ge_balance_positive_cancel_right_product_secondoutputreal) = 0) /\ (ge_balance_negative_cancel_right_product_secondoutputreal) = S ge_signed_half_cancel_right_product_secondoutputrealdecode))) /\ ((((((((ge_first_rp_cancel_right_product_second) * (ge_second_rp_cancel_right_product_second))) + (((ge_first_rn_cancel_right_product_second) * (ge_second_rn_cancel_right_product_second))))) + (((((ge_first_ip_cancel_right_product_second) * (ge_second_in_cancel_right_product_second))) + (((ge_first_in_cancel_right_product_second) * (ge_second_ip_cancel_right_product_second))))))) + ge_balance_negative_cancel_right_product_secondoutputreal = (((((((ge_first_rp_cancel_right_product_second) * (ge_second_rn_cancel_right_product_second))) + (((ge_first_rn_cancel_right_product_second) * (ge_second_rp_cancel_right_product_second))))) + (((((ge_first_ip_cancel_right_product_second) * (ge_second_ip_cancel_right_product_second))) + (((ge_first_in_cancel_right_product_second) * (ge_second_in_cancel_right_product_second))))))) + ge_balance_positive_cancel_right_product_secondoutputreal))) /\ (exists ge_balance_positive_cancel_right_product_secondoutputimaginary ge_balance_negative_cancel_right_product_secondoutputimaginary. (((((ge_representation_imaginary_code_cancel_right_product_secondoutput) = 2 * (ge_balance_positive_cancel_right_product_secondoutputimaginary) /\ (ge_balance_negative_cancel_right_product_secondoutputimaginary) = 0) \/ exists ge_signed_half_cancel_right_product_secondoutputimaginarydecode. (((ge_representation_imaginary_code_cancel_right_product_secondoutput) = 2 * ge_signed_half_cancel_right_product_secondoutputimaginarydecode + 1 /\ (ge_balance_positive_cancel_right_product_secondoutputimaginary) = 0) /\ (ge_balance_negative_cancel_right_product_secondoutputimaginary) = S ge_signed_half_cancel_right_product_secondoutputimaginarydecode))) /\ ((((((((ge_first_rp_cancel_right_product_second) * (ge_second_ip_cancel_right_product_second))) + (((ge_first_rn_cancel_right_product_second) * (ge_second_in_cancel_right_product_second))))) + (((((ge_first_ip_cancel_right_product_second) * (ge_second_rp_cancel_right_product_second))) + (((ge_first_in_cancel_right_product_second) * (ge_second_rn_cancel_right_product_second))))))) + ge_balance_negative_cancel_right_product_secondoutputimaginary = (((((((ge_first_rp_cancel_right_product_second) * (ge_second_in_cancel_right_product_second))) + (((ge_first_rn_cancel_right_product_second) * (ge_second_ip_cancel_right_product_second))))) + (((((ge_first_ip_cancel_right_product_second) * (ge_second_rn_cancel_right_product_second))) + (((ge_first_in_cancel_right_product_second) * (ge_second_rp_cancel_right_product_second))))))) + ge_balance_positive_cancel_right_product_secondoutputimaginary))))))))) -> a=b

Complete tactic proof in conservative notation

All 23 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

23 script commands · 3 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.

Named ingredients (2)
gaussian multiply cancel leftgaussian multiply commutative
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 t
  5. L5
    intro hc
  6. L6
    intro hA
  7. L7
    intro hB
02Use earlier factsL8–17

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

  1. L8
    specialize gaussian_multiply_cancel_left (c)
  2. L9
    specialize gaussian_multiply_cancel_left (a)
  3. L10
    specialize gaussian_multiply_cancel_left (b)
  4. L11
    specialize gaussian_multiply_cancel_left (t)
  5. L12
    apply gaussian_multiply_cancel_left
  6. L13
    exact hc
  7. L14
    specialize gaussian_multiply_commutative (a)
  8. L15
    specialize gaussian_multiply_commutative (c)
  9. L16
    specialize gaussian_multiply_commutative (t)
  10. L17
    apply gaussian_multiply_commutative
03Use earlier factsL18–23

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

  1. L18
    exact hA
  2. L19
    specialize gaussian_multiply_commutative (b)
  3. L20
    specialize gaussian_multiply_commutative (c)
  4. L21
    specialize gaussian_multiply_commutative (t)
  5. L22
    apply gaussian_multiply_commutative
  6. L23
    exact hB

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro t
  5. 0005intro hc
  6. 0006intro hA
  7. 0007intro hB
  8. 0008specialize gaussian_multiply_cancel_left (c)
  9. 0009specialize gaussian_multiply_cancel_left (a)
  10. 0010specialize gaussian_multiply_cancel_left (b)
  11. 0011specialize gaussian_multiply_cancel_left (t)
  12. 0012apply gaussian_multiply_cancel_left
  13. 0013exact hc
  14. 0014specialize gaussian_multiply_commutative (a)
  15. 0015specialize gaussian_multiply_commutative (c)
  16. 0016specialize gaussian_multiply_commutative (t)
  17. 0017apply gaussian_multiply_commutative
  18. 0018exact hA
  19. 0019specialize gaussian_multiply_commutative (b)
  20. 0020specialize gaussian_multiply_commutative (c)
  21. 0021specialize gaussian_multiply_commutative (t)
  22. 0022apply gaussian_multiply_commutative
  23. 0023exact hB