GC0001

crt_mod_one_universal

Every pair of natural residues is constructively congruent modulo one.

Alpha v34 checked-use · first admitted v25 · 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.

Historical partial components only: this chapter proves canonical solutions under successive-merge compatibility and in the pairwise-compatible dominating-last case. G011 is now closed in the separate Alpha-v27 generalized-crt branch for arbitrary pairwise-compatible finite lists, including noncoprime moduli. Full G011 proof · Alpha v27

Exact theorem in conservative defined notation

∀ a. ∀ b. ModEq(1,a,b)

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

Definition DAG

ModEq(m,a,b) · 1

Actual proof prerequisites

one_mul · checked external prerequisiteadd_comm · checked external prerequisite
Original expanded first-order 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)

Complete unchanged native tactic proof

All 14 lines are the exact independently kernel-checked original script.

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.

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 assumptionsL1–2

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 defined 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