CG000B

linear_congruence_modulus_one_bounded_iff_zero

For modulus one the unique strictly bounded solution is zero for every coefficient and target.

Alpha v34 checked-use · first admitted v34 · 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.

Each statement retains its explicit modulus, coprimality and divisibility assumptions. These twelve arithmetic laws do not assert all order, primitive-root, Carmichael, exponential or simultaneous-polynomial congruence goals are finished.

Exact theorem in conservative defined notation

∀ a. ∀ b. ∀ x. (Lt(x,1) ∧ ModEq(1,a · x,b) → x = 0) ∧ (x = 0 → Lt(x,1) ∧ ModEq(1,a · x,b))

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

Definition DAG

Lt(a,b) · 2ModEq(m,a,b) · 2

Actual proof prerequisites

le_zero · checked external prerequisitele_of_succ_le_succ · checked external prerequisitezero_add · checked external prerequisitecrt_mod_one_universal · checked external prerequisite
Original expanded first-order statement 
forall a b x. ((((((exists lcc_gap_one_bound. lcc_gap_one_bound+S (x)=(1)) /\ ((exists hgcrt_mod_left_lcc_one_sol hgcrt_mod_right_lcc_one_sol. (a * x) + 1 * hgcrt_mod_left_lcc_one_sol = b + 1 * hgcrt_mod_right_lcc_one_sol)))) -> (x=0)) /\ (((x=0) -> (((exists lcc_gap_one_bound. lcc_gap_one_bound+S (x)=(1)) /\ ((exists hgcrt_mod_left_lcc_one_sol hgcrt_mod_right_lcc_one_sol. (a * x) + 1 * hgcrt_mod_left_lcc_one_sol = b + 1 * hgcrt_mod_right_lcc_one_sol))))))))

Complete tactic proof in conservative notation

All 20 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

20 script commands · 10 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 assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro x
02Separate the logical casesL4–4

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

  1. L4
    split
03Fix variables and assumptionsL5–5

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

  1. L5
    intro h
04Separate the logical casesL6–6

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

  1. L6
    cases h
05Use earlier factsL7–12

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

  1. L7
    specialize le_zero (x)
  2. L8
    apply le_zero
  3. L9
    specialize le_of_succ_le_succ (x)
  4. L10
    specialize le_of_succ_le_succ (0)
  5. L11
    apply le_of_succ_le_succ
  6. L12
    exact h_left
06Fix variables and assumptionsL13–13

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

  1. L13
    intro h
07Separate the logical casesL14–14

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

  1. L14
    split
08Calculate and transport equalitiesL15–15

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

  1. L15
    rewrite h
09Construct an explicit witnessL16–16

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

  1. L16
    exists 0
10Use earlier factsL17–20

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

  1. L17
    apply zero_add
  2. L18
    specialize crt_mod_one_universal (a*x)
  3. L19
    specialize crt_mod_one_universal (b)
  4. L20
    apply crt_mod_one_universal

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro x
  4. 0004split
  5. 0005intro h
  6. 0006cases h
  7. 0007specialize le_zero (x)
  8. 0008apply le_zero
  9. 0009specialize le_of_succ_le_succ (x)
  10. 0010specialize le_of_succ_le_succ (0)
  11. 0011apply le_of_succ_le_succ
  12. 0012exact h_left
  13. 0013intro h
  14. 0014split
  15. 0015rewrite h
  16. 0016exists 0
  17. 0017apply zero_add
  18. 0018specialize crt_mod_one_universal (a*x)
  19. 0019specialize crt_mod_one_universal (b)
  20. 0020apply crt_mod_one_universal