GF0015

gaussian_norm_value_transport

Equality transports the uniquely defined actual Gaussian norm value.

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

∀ z. ∀ N. ∀ M. N = M → GNorm(z,N) → GNorm(z,M)

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

Definition DAG

GNorm(z,n) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall z N M. N=M -> (exists ge_norm_rp_norm_old_value ge_norm_rn_norm_old_value ge_norm_ip_norm_old_value ge_norm_in_norm_old_value. ((exists ge_representation_real_code_norm_old_valuerepresentation ge_representation_imaginary_code_norm_old_valuerepresentation. (((z) = ((ge_representation_real_code_norm_old_valuerepresentation) + (ge_representation_imaginary_code_norm_old_valuerepresentation)) * S ((ge_representation_real_code_norm_old_valuerepresentation) + (ge_representation_imaginary_code_norm_old_valuerepresentation)) + ((ge_representation_imaginary_code_norm_old_valuerepresentation) + (ge_representation_imaginary_code_norm_old_valuerepresentation))) /\ ((exists ge_balance_positive_norm_old_valuerepresentationreal ge_balance_negative_norm_old_valuerepresentationreal. (((((ge_representation_real_code_norm_old_valuerepresentation) = 2 * (ge_balance_positive_norm_old_valuerepresentationreal) /\ (ge_balance_negative_norm_old_valuerepresentationreal) = 0) \/ exists ge_signed_half_norm_old_valuerepresentationrealdecode. (((ge_representation_real_code_norm_old_valuerepresentation) = 2 * ge_signed_half_norm_old_valuerepresentationrealdecode + 1 /\ (ge_balance_positive_norm_old_valuerepresentationreal) = 0) /\ (ge_balance_negative_norm_old_valuerepresentationreal) = S ge_signed_half_norm_old_valuerepresentationrealdecode))) /\ ((ge_norm_rp_norm_old_value) + ge_balance_negative_norm_old_valuerepresentationreal = (ge_norm_rn_norm_old_value) + ge_balance_positive_norm_old_valuerepresentationreal))) /\ (exists ge_balance_positive_norm_old_valuerepresentationimaginary ge_balance_negative_norm_old_valuerepresentationimaginary. (((((ge_representation_imaginary_code_norm_old_valuerepresentation) = 2 * (ge_balance_positive_norm_old_valuerepresentationimaginary) /\ (ge_balance_negative_norm_old_valuerepresentationimaginary) = 0) \/ exists ge_signed_half_norm_old_valuerepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_old_valuerepresentation) = 2 * ge_signed_half_norm_old_valuerepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_old_valuerepresentationimaginary) = 0) /\ (ge_balance_negative_norm_old_valuerepresentationimaginary) = S ge_signed_half_norm_old_valuerepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_old_value) + ge_balance_negative_norm_old_valuerepresentationimaginary = (ge_norm_in_norm_old_value) + ge_balance_positive_norm_old_valuerepresentationimaginary)))))) /\ (exists ge_real_square_norm_old_valuesquare ge_imaginary_square_norm_old_valuesquare. ((((((ge_norm_rp_norm_old_value) * (ge_norm_rp_norm_old_value))) + (((ge_norm_rn_norm_old_value) * (ge_norm_rn_norm_old_value)))) = ((ge_real_square_norm_old_valuesquare) + (((((ge_norm_rp_norm_old_value) * (ge_norm_rn_norm_old_value))) + (((ge_norm_rn_norm_old_value) * (ge_norm_rp_norm_old_value))))))) /\ ((((((ge_norm_ip_norm_old_value) * (ge_norm_ip_norm_old_value))) + (((ge_norm_in_norm_old_value) * (ge_norm_in_norm_old_value)))) = ((ge_imaginary_square_norm_old_valuesquare) + (((((ge_norm_ip_norm_old_value) * (ge_norm_in_norm_old_value))) + (((ge_norm_in_norm_old_value) * (ge_norm_ip_norm_old_value))))))) /\ ((N) = ge_real_square_norm_old_valuesquare + ge_imaginary_square_norm_old_valuesquare)))))) -> (exists ge_norm_rp_norm_new_value ge_norm_rn_norm_new_value ge_norm_ip_norm_new_value ge_norm_in_norm_new_value. ((exists ge_representation_real_code_norm_new_valuerepresentation ge_representation_imaginary_code_norm_new_valuerepresentation. (((z) = ((ge_representation_real_code_norm_new_valuerepresentation) + (ge_representation_imaginary_code_norm_new_valuerepresentation)) * S ((ge_representation_real_code_norm_new_valuerepresentation) + (ge_representation_imaginary_code_norm_new_valuerepresentation)) + ((ge_representation_imaginary_code_norm_new_valuerepresentation) + (ge_representation_imaginary_code_norm_new_valuerepresentation))) /\ ((exists ge_balance_positive_norm_new_valuerepresentationreal ge_balance_negative_norm_new_valuerepresentationreal. (((((ge_representation_real_code_norm_new_valuerepresentation) = 2 * (ge_balance_positive_norm_new_valuerepresentationreal) /\ (ge_balance_negative_norm_new_valuerepresentationreal) = 0) \/ exists ge_signed_half_norm_new_valuerepresentationrealdecode. (((ge_representation_real_code_norm_new_valuerepresentation) = 2 * ge_signed_half_norm_new_valuerepresentationrealdecode + 1 /\ (ge_balance_positive_norm_new_valuerepresentationreal) = 0) /\ (ge_balance_negative_norm_new_valuerepresentationreal) = S ge_signed_half_norm_new_valuerepresentationrealdecode))) /\ ((ge_norm_rp_norm_new_value) + ge_balance_negative_norm_new_valuerepresentationreal = (ge_norm_rn_norm_new_value) + ge_balance_positive_norm_new_valuerepresentationreal))) /\ (exists ge_balance_positive_norm_new_valuerepresentationimaginary ge_balance_negative_norm_new_valuerepresentationimaginary. (((((ge_representation_imaginary_code_norm_new_valuerepresentation) = 2 * (ge_balance_positive_norm_new_valuerepresentationimaginary) /\ (ge_balance_negative_norm_new_valuerepresentationimaginary) = 0) \/ exists ge_signed_half_norm_new_valuerepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_new_valuerepresentation) = 2 * ge_signed_half_norm_new_valuerepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_new_valuerepresentationimaginary) = 0) /\ (ge_balance_negative_norm_new_valuerepresentationimaginary) = S ge_signed_half_norm_new_valuerepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_new_value) + ge_balance_negative_norm_new_valuerepresentationimaginary = (ge_norm_in_norm_new_value) + ge_balance_positive_norm_new_valuerepresentationimaginary)))))) /\ (exists ge_real_square_norm_new_valuesquare ge_imaginary_square_norm_new_valuesquare. ((((((ge_norm_rp_norm_new_value) * (ge_norm_rp_norm_new_value))) + (((ge_norm_rn_norm_new_value) * (ge_norm_rn_norm_new_value)))) = ((ge_real_square_norm_new_valuesquare) + (((((ge_norm_rp_norm_new_value) * (ge_norm_rn_norm_new_value))) + (((ge_norm_rn_norm_new_value) * (ge_norm_rp_norm_new_value))))))) /\ ((((((ge_norm_ip_norm_new_value) * (ge_norm_ip_norm_new_value))) + (((ge_norm_in_norm_new_value) * (ge_norm_in_norm_new_value)))) = ((ge_imaginary_square_norm_new_valuesquare) + (((((ge_norm_ip_norm_new_value) * (ge_norm_in_norm_new_value))) + (((ge_norm_in_norm_new_value) * (ge_norm_ip_norm_new_value))))))) /\ ((M) = ge_real_square_norm_new_valuesquare + ge_imaginary_square_norm_new_valuesquare))))))

Complete tactic proof in conservative notation

All 7 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

7 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro z
  2. L2
    intro N
  3. L3
    intro M
  4. L4
    intro heq
  5. L5
    intro h
02Calculate and transport equalitiesL6–6

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

  1. L6
    rewrite heq at h
03Use earlier factsL7–7

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

  1. L7
    exact h

Library-wide reading audit

Original defined command ledger · 7 lines
  1. 0001intro z
  2. 0002intro N
  3. 0003intro M
  4. 0004intro heq
  5. 0005intro h
  6. 0006rewrite heq at h
  7. 0007exact h