CG000A

linear_congruence_zero_modulus_zero_coefficient_iff

With coefficient and modulus both zero, every natural x is a solution exactly when the target is zero.

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

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

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

Definition DAG

ModEq(m,a,b) · 4

Actual proof prerequisites

mul_zero_left · checked external prerequisitemod_eq_zero_iff_eq · checked external prerequisite
Original expanded first-order statement 
forall b x. ((((exists hgcrt_mod_left_lcc_zero_coefficient hgcrt_mod_right_lcc_zero_coefficient. (0 * x) + 0 * hgcrt_mod_left_lcc_zero_coefficient = b + 0 * hgcrt_mod_right_lcc_zero_coefficient) -> (b=0)) /\ (((b=0) -> (exists hgcrt_mod_left_lcc_zero_coefficient hgcrt_mod_right_lcc_zero_coefficient. (0 * x) + 0 * hgcrt_mod_left_lcc_zero_coefficient = b + 0 * hgcrt_mod_right_lcc_zero_coefficient)))))

Complete tactic proof in conservative notation

All 23 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

23 script commands · 13 reading checkpoints · 2 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 b
  2. L2
    intro x
02Establish h0L3–4

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul zero left.

  1. L3
    have h0 : 0*x=0
  2. L4
    apply mul_zero_left
03Establish hcL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq zero iff eq.

  1. L5
    have hc : (ModEq(0,0 · x,b) → 0 · x = b) ∧ (0 · x = b → ModEq(0,0 · x,b))Definitions: ModEq(0,0 · x,b)Original native command in the exact edition
  2. L6
    specialize mod_eq_zero_iff_eq (0*x)
  3. L7
    specialize mod_eq_zero_iff_eq (b)
  4. L8
    apply mod_eq_zero_iff_eq
04Separate the logical casesL9–10

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

  1. L9
    cases hc
  2. L10
    split
05Fix variables and assumptionsL11–11

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

  1. L11
    intro h
06Calculate and transport equalitiesL12–14

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

  1. L12
    symm
  2. L13
    trans 0*x
  3. L14
    symm
07Use earlier factsL15–17

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

  1. L15
    exact h0
  2. L16
    apply hc_left
  3. L17
    exact h
08Fix variables and assumptionsL18–18

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

  1. L18
    intro h
09Use earlier factsL19–19

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

  1. L19
    apply hc_right
10Calculate and transport equalitiesL20–20

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

  1. L20
    trans 0
11Use earlier factsL21–21

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

  1. L21
    exact h0
12Calculate and transport equalitiesL22–22

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

  1. L22
    symm
13Use earlier factsL23–23

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

  1. L23
    exact h

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro b
  2. 0002intro x
  3. 0003have h0 : 0*x=0
  4. 0004apply mul_zero_left
  5. 0005have hc : (ModEq(0,0 · x,b) → 0 · x = b) ∧ (0 · x = b → ModEq(0,0 · x,b))
  6. 0006specialize mod_eq_zero_iff_eq (0*x)
  7. 0007specialize mod_eq_zero_iff_eq (b)
  8. 0008apply mod_eq_zero_iff_eq
  9. 0009cases hc
  10. 0010split
  11. 0011intro h
  12. 0012symm
  13. 0013trans 0*x
  14. 0014symm
  15. 0015exact h0
  16. 0016apply hc_left
  17. 0017exact h
  18. 0018intro h
  19. 0019apply hc_right
  20. 0020trans 0
  21. 0021exact h0
  22. 0022symm
  23. 0023exact h