GF0015

gaussian_norm_value_transport

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

Equality transports the uniquely defined actual Gaussian norm value.

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 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))))))

Constructive proof overview

Generated structural guide

Equality transports the uniquely defined actual Gaussian norm value.

The unchanged tactic script uses 0 declared prerequisites and contains 7 exact native proof lines.

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

Proof neighborhood

Direct dependencies

none

Direct dependents

GF0019 gaussian_unit_has_norm_one GF001B gaussian_unit_iff_norm_one GF001C gaussian_unit_decidable GF001F gaussian_multiply_zero_implies_zero_factor GF003F gaussian_unit_product GF0052 gaussian_division_divisible_remainder_zero GF0059 gaussian_associate_norm GF0064 gaussian_gcd_bezout_bounded_exists GF0079 gaussian_search_nonunit_norm_two GF0080 gaussian_irreducible_divisor_bounded_norm GF0094 gaussian_irreducible_factorization_bounded_norm

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

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.

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 exact 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