GC0001

crt_mod_one_universal

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

Every pair of natural residues is constructively congruent modulo one.

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 a b. (exists hgcrt_mod_left_gcrt_gcomp_one hgcrt_mod_right_gcrt_gcomp_one. a + 1 * hgcrt_mod_left_gcrt_gcomp_one = b + 1 * hgcrt_mod_right_gcrt_gcomp_one)

Constructive proof overview

Generated structural guide

Every pair of natural residues is constructively congruent modulo one.

The unchanged tactic script uses 2 declared prerequisites and contains 14 exact native proof lines.

Alpha v34 checked-use · first admitted v25 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

one_mul Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

none

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

14 script commands · 8 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 assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Construct an explicit witnessL3–4

Supply the displayed value, then prove that it has the required property.

  1. L3
    exists b
  2. L4
    exists a
03Calculate and transport equalitiesL5–7

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

  1. L5
    trans a + b
  2. L6
    congr
  3. L7
    refl
04Use earlier factsL8–8

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

  1. L8
    apply one_mul
05Calculate and transport equalitiesL9–9

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

  1. L9
    trans b + a
06Use earlier factsL10–10

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

  1. L10
    apply add_comm
07Calculate and transport equalitiesL11–13

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

  1. L11
    congr
  2. L12
    refl
  3. L13
    symm
08Use earlier factsL14–14

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

  1. L14
    apply one_mul

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003exists b
  4. 0004exists a
  5. 0005trans a + b
  6. 0006congr
  7. 0007refl
  8. 0008apply one_mul
  9. 0009trans b + a
  10. 0010apply add_comm
  11. 0011congr
  12. 0012refl
  13. 0013symm
  14. 0014apply one_mul

Separate complete second-wave branches: Full G011 proof · Alpha v27.