CG0006

linear_congruence_bounded_parameter_unique

The actual progression parameter is unique for nonzero M, even without imposing a redundant parameter bound.

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

∀ M. ∀ r. ∀ x. ∀ t. ∀ u. ¬M = 0 → x = r + M · t → x = r + M · u → t = u

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

Definition DAG

none

Actual proof prerequisites

mul_left_cancel_nonzero · checked external prerequisiteadd_left_cancel · checked external prerequisite
Original expanded first-order statement 
forall M r x t u. (~(M=0)) -> (x=r+M*t) -> (x=r+M*u) -> (t=u)

Complete tactic proof in conservative notation

All 21 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

21 script commands · 4 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 assumptionsL1–8

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

  1. L1
    intro M
  2. L2
    intro r
  3. L3
    intro x
  4. L4
    intro t
  5. L5
    intro u
  6. L6
    intro hM
  7. L7
    intro ht
  8. L8
    intro hu
02Use earlier factsL9–17

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

  1. L9
    specialize mul_left_cancel_nonzero (M)
  2. L10
    specialize mul_left_cancel_nonzero (t)
  3. L11
    specialize mul_left_cancel_nonzero (u)
  4. L12
    apply mul_left_cancel_nonzero
  5. L13
    exact hM
  6. L14
    specialize add_left_cancel (r)
  7. L15
    specialize add_left_cancel (M*t)
  8. L16
    specialize add_left_cancel (M*u)
  9. L17
    apply add_left_cancel
03Calculate and transport equalitiesL18–19

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

  1. L18
    trans x
  2. L19
    symm
04Use earlier factsL20–21

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

  1. L20
    exact ht
  2. L21
    exact hu

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro M
  2. 0002intro r
  3. 0003intro x
  4. 0004intro t
  5. 0005intro u
  6. 0006intro hM
  7. 0007intro ht
  8. 0008intro hu
  9. 0009specialize mul_left_cancel_nonzero (M)
  10. 0010specialize mul_left_cancel_nonzero (t)
  11. 0011specialize mul_left_cancel_nonzero (u)
  12. 0012apply mul_left_cancel_nonzero
  13. 0013exact hM
  14. 0014specialize add_left_cancel (r)
  15. 0015specialize add_left_cancel (M*t)
  16. 0016specialize add_left_cancel (M*u)
  17. 0017apply add_left_cancel
  18. 0018trans x
  19. 0019symm
  20. 0020exact ht
  21. 0021exact hu