GF0042

gaussian_divides_input_valid

A witnessed Gaussian divisor belongs to the actual canonical Gaussian carrier.

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

∀ d. ∀ z. GDvd(d,z) → ZPairValid(d)

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

Definition DAG

ZPairValid(z) · 1GDvd(d,z) · 1

Actual proof prerequisites

gaussian_multiply_input_left_valid
Original expanded first-order statement 
forall d z. (exists gr_quotient_divisor_domain. (exists ge_first_rp_divisor_domainproduct ge_first_rn_divisor_domainproduct ge_first_ip_divisor_domainproduct ge_first_in_divisor_domainproduct ge_second_rp_divisor_domainproduct ge_second_rn_divisor_domainproduct ge_second_ip_divisor_domainproduct ge_second_in_divisor_domainproduct. ((exists ge_representation_real_code_divisor_domainproductfirst ge_representation_imaginary_code_divisor_domainproductfirst. (((d) = ((ge_representation_real_code_divisor_domainproductfirst) + (ge_representation_imaginary_code_divisor_domainproductfirst)) * S ((ge_representation_real_code_divisor_domainproductfirst) + (ge_representation_imaginary_code_divisor_domainproductfirst)) + ((ge_representation_imaginary_code_divisor_domainproductfirst) + (ge_representation_imaginary_code_divisor_domainproductfirst))) /\ ((exists ge_balance_positive_divisor_domainproductfirstreal ge_balance_negative_divisor_domainproductfirstreal. (((((ge_representation_real_code_divisor_domainproductfirst) = 2 * (ge_balance_positive_divisor_domainproductfirstreal) /\ (ge_balance_negative_divisor_domainproductfirstreal) = 0) \/ exists ge_signed_half_divisor_domainproductfirstrealdecode. (((ge_representation_real_code_divisor_domainproductfirst) = 2 * ge_signed_half_divisor_domainproductfirstrealdecode + 1 /\ (ge_balance_positive_divisor_domainproductfirstreal) = 0) /\ (ge_balance_negative_divisor_domainproductfirstreal) = S ge_signed_half_divisor_domainproductfirstrealdecode))) /\ ((ge_first_rp_divisor_domainproduct) + ge_balance_negative_divisor_domainproductfirstreal = (ge_first_rn_divisor_domainproduct) + ge_balance_positive_divisor_domainproductfirstreal))) /\ (exists ge_balance_positive_divisor_domainproductfirstimaginary ge_balance_negative_divisor_domainproductfirstimaginary. (((((ge_representation_imaginary_code_divisor_domainproductfirst) = 2 * (ge_balance_positive_divisor_domainproductfirstimaginary) /\ (ge_balance_negative_divisor_domainproductfirstimaginary) = 0) \/ exists ge_signed_half_divisor_domainproductfirstimaginarydecode. (((ge_representation_imaginary_code_divisor_domainproductfirst) = 2 * ge_signed_half_divisor_domainproductfirstimaginarydecode + 1 /\ (ge_balance_positive_divisor_domainproductfirstimaginary) = 0) /\ (ge_balance_negative_divisor_domainproductfirstimaginary) = S ge_signed_half_divisor_domainproductfirstimaginarydecode))) /\ ((ge_first_ip_divisor_domainproduct) + ge_balance_negative_divisor_domainproductfirstimaginary = (ge_first_in_divisor_domainproduct) + ge_balance_positive_divisor_domainproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_divisor_domainproductsecond ge_representation_imaginary_code_divisor_domainproductsecond. (((gr_quotient_divisor_domain) = ((ge_representation_real_code_divisor_domainproductsecond) + (ge_representation_imaginary_code_divisor_domainproductsecond)) * S ((ge_representation_real_code_divisor_domainproductsecond) + (ge_representation_imaginary_code_divisor_domainproductsecond)) + ((ge_representation_imaginary_code_divisor_domainproductsecond) + (ge_representation_imaginary_code_divisor_domainproductsecond))) /\ ((exists ge_balance_positive_divisor_domainproductsecondreal ge_balance_negative_divisor_domainproductsecondreal. (((((ge_representation_real_code_divisor_domainproductsecond) = 2 * (ge_balance_positive_divisor_domainproductsecondreal) /\ (ge_balance_negative_divisor_domainproductsecondreal) = 0) \/ exists ge_signed_half_divisor_domainproductsecondrealdecode. (((ge_representation_real_code_divisor_domainproductsecond) = 2 * ge_signed_half_divisor_domainproductsecondrealdecode + 1 /\ (ge_balance_positive_divisor_domainproductsecondreal) = 0) /\ (ge_balance_negative_divisor_domainproductsecondreal) = S ge_signed_half_divisor_domainproductsecondrealdecode))) /\ ((ge_second_rp_divisor_domainproduct) + ge_balance_negative_divisor_domainproductsecondreal = (ge_second_rn_divisor_domainproduct) + ge_balance_positive_divisor_domainproductsecondreal))) /\ (exists ge_balance_positive_divisor_domainproductsecondimaginary ge_balance_negative_divisor_domainproductsecondimaginary. (((((ge_representation_imaginary_code_divisor_domainproductsecond) = 2 * (ge_balance_positive_divisor_domainproductsecondimaginary) /\ (ge_balance_negative_divisor_domainproductsecondimaginary) = 0) \/ exists ge_signed_half_divisor_domainproductsecondimaginarydecode. (((ge_representation_imaginary_code_divisor_domainproductsecond) = 2 * ge_signed_half_divisor_domainproductsecondimaginarydecode + 1 /\ (ge_balance_positive_divisor_domainproductsecondimaginary) = 0) /\ (ge_balance_negative_divisor_domainproductsecondimaginary) = S ge_signed_half_divisor_domainproductsecondimaginarydecode))) /\ ((ge_second_ip_divisor_domainproduct) + ge_balance_negative_divisor_domainproductsecondimaginary = (ge_second_in_divisor_domainproduct) + ge_balance_positive_divisor_domainproductsecondimaginary)))))) /\ (exists ge_representation_real_code_divisor_domainproductoutput ge_representation_imaginary_code_divisor_domainproductoutput. (((z) = ((ge_representation_real_code_divisor_domainproductoutput) + (ge_representation_imaginary_code_divisor_domainproductoutput)) * S ((ge_representation_real_code_divisor_domainproductoutput) + (ge_representation_imaginary_code_divisor_domainproductoutput)) + ((ge_representation_imaginary_code_divisor_domainproductoutput) + (ge_representation_imaginary_code_divisor_domainproductoutput))) /\ ((exists ge_balance_positive_divisor_domainproductoutputreal ge_balance_negative_divisor_domainproductoutputreal. (((((ge_representation_real_code_divisor_domainproductoutput) = 2 * (ge_balance_positive_divisor_domainproductoutputreal) /\ (ge_balance_negative_divisor_domainproductoutputreal) = 0) \/ exists ge_signed_half_divisor_domainproductoutputrealdecode. (((ge_representation_real_code_divisor_domainproductoutput) = 2 * ge_signed_half_divisor_domainproductoutputrealdecode + 1 /\ (ge_balance_positive_divisor_domainproductoutputreal) = 0) /\ (ge_balance_negative_divisor_domainproductoutputreal) = S ge_signed_half_divisor_domainproductoutputrealdecode))) /\ ((((((((ge_first_rp_divisor_domainproduct) * (ge_second_rp_divisor_domainproduct))) + (((ge_first_rn_divisor_domainproduct) * (ge_second_rn_divisor_domainproduct))))) + (((((ge_first_ip_divisor_domainproduct) * (ge_second_in_divisor_domainproduct))) + (((ge_first_in_divisor_domainproduct) * (ge_second_ip_divisor_domainproduct))))))) + ge_balance_negative_divisor_domainproductoutputreal = (((((((ge_first_rp_divisor_domainproduct) * (ge_second_rn_divisor_domainproduct))) + (((ge_first_rn_divisor_domainproduct) * (ge_second_rp_divisor_domainproduct))))) + (((((ge_first_ip_divisor_domainproduct) * (ge_second_ip_divisor_domainproduct))) + (((ge_first_in_divisor_domainproduct) * (ge_second_in_divisor_domainproduct))))))) + ge_balance_positive_divisor_domainproductoutputreal))) /\ (exists ge_balance_positive_divisor_domainproductoutputimaginary ge_balance_negative_divisor_domainproductoutputimaginary. (((((ge_representation_imaginary_code_divisor_domainproductoutput) = 2 * (ge_balance_positive_divisor_domainproductoutputimaginary) /\ (ge_balance_negative_divisor_domainproductoutputimaginary) = 0) \/ exists ge_signed_half_divisor_domainproductoutputimaginarydecode. (((ge_representation_imaginary_code_divisor_domainproductoutput) = 2 * ge_signed_half_divisor_domainproductoutputimaginarydecode + 1 /\ (ge_balance_positive_divisor_domainproductoutputimaginary) = 0) /\ (ge_balance_negative_divisor_domainproductoutputimaginary) = S ge_signed_half_divisor_domainproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_divisor_domainproduct) * (ge_second_ip_divisor_domainproduct))) + (((ge_first_rn_divisor_domainproduct) * (ge_second_in_divisor_domainproduct))))) + (((((ge_first_ip_divisor_domainproduct) * (ge_second_rp_divisor_domainproduct))) + (((ge_first_in_divisor_domainproduct) * (ge_second_rn_divisor_domainproduct))))))) + ge_balance_negative_divisor_domainproductoutputimaginary = (((((((ge_first_rp_divisor_domainproduct) * (ge_second_in_divisor_domainproduct))) + (((ge_first_rn_divisor_domainproduct) * (ge_second_ip_divisor_domainproduct))))) + (((((ge_first_ip_divisor_domainproduct) * (ge_second_rn_divisor_domainproduct))) + (((ge_first_in_divisor_domainproduct) * (ge_second_rp_divisor_domainproduct))))))) + ge_balance_positive_divisor_domainproductoutputimaginary)))))))))) -> (exists ge_real_positive_divisor_valid ge_real_negative_divisor_valid ge_imaginary_positive_divisor_valid ge_imaginary_negative_divisor_valid. (exists ge_real_code_divisor_validdecode ge_imaginary_code_divisor_validdecode. (((d) = ((ge_real_code_divisor_validdecode) + (ge_imaginary_code_divisor_validdecode)) * S ((ge_real_code_divisor_validdecode) + (ge_imaginary_code_divisor_validdecode)) + ((ge_imaginary_code_divisor_validdecode) + (ge_imaginary_code_divisor_validdecode))) /\ (((((ge_real_code_divisor_validdecode) = 2 * (ge_real_positive_divisor_valid) /\ (ge_real_negative_divisor_valid) = 0) \/ exists ge_signed_half_ge_divisor_validdecode_real. (((ge_real_code_divisor_validdecode) = 2 * ge_signed_half_ge_divisor_validdecode_real + 1 /\ (ge_real_positive_divisor_valid) = 0) /\ (ge_real_negative_divisor_valid) = S ge_signed_half_ge_divisor_validdecode_real))) /\ ((((ge_imaginary_code_divisor_validdecode) = 2 * (ge_imaginary_positive_divisor_valid) /\ (ge_imaginary_negative_divisor_valid) = 0) \/ exists ge_signed_half_ge_divisor_validdecode_imaginary. (((ge_imaginary_code_divisor_validdecode) = 2 * ge_signed_half_ge_divisor_validdecode_imaginary + 1 /\ (ge_imaginary_positive_divisor_valid) = 0) /\ (ge_imaginary_negative_divisor_valid) = S ge_signed_half_ge_divisor_validdecode_imaginary)))))))

Complete tactic proof in conservative notation

All 9 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

9 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 (1)
gaussian multiply input left valid
01Fix variables and assumptionsL1–3

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

  1. L1
    intro d
  2. L2
    intro z
  3. L3
    intro h
02Separate the logical casesL4–4

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

  1. L4
    cases h
03Use earlier factsL5–9

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

  1. L5
    specialize gaussian_multiply_input_left_valid (d)
  2. L6
    specialize gaussian_multiply_input_left_valid (x)
  3. L7
    specialize gaussian_multiply_input_left_valid (z)
  4. L8
    apply gaussian_multiply_input_left_valid
  5. L9
    exact h_witness

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro d
  2. 0002intro z
  3. 0003intro h
  4. 0004cases h
  5. 0005specialize gaussian_multiply_input_left_valid (d)
  6. 0006specialize gaussian_multiply_input_left_valid (x)
  7. 0007specialize gaussian_multiply_input_left_valid (z)
  8. 0008apply gaussian_multiply_input_left_valid
  9. 0009exact h_witness