BA0007

cf_approximation_unit_determinant_coprime

Every actual determinant-one adjacent pair has a coprime current numerator and denominator, including 0/1.

Alpha v34 checked-use · first admitted v29 · 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.

The initial 0/1 convergent is included: u is natural, not necessarily positive. Comparison denominators are strictly smaller and positive. Signed competitors are represented by an arbitrary difference rp−rn. Approximation inequalities are proved from the trace, never stored as assumptions in Convergent.

Exact theorem in conservative defined notation

∀ u. ∀ U. ∀ v. ∀ V. u · V + 1 = U · v ∨ U · v + 1 = u · V → ∀ x. (∃ y. u = x · y) → (∃ y. v = x · y) → x = 1

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

Definition DAG

none

Actual proof prerequisites

balanced_bezout_one_implies_coprime · checked external prerequisitemul_comm · checked external prerequisitezero_add · checked external prerequisiteadd_succ_left · checked external prerequisite
Original expanded first-order statement 
forall u U v V. (u * V + 1 = U * v \/ U * v + 1 = u * V) -> (forall frp_divisor_cfba_determinant_coprime. (exists frp_left_factor_cfba_determinant_coprime. u = frp_divisor_cfba_determinant_coprime * frp_left_factor_cfba_determinant_coprime) -> (exists frp_right_factor_cfba_determinant_coprime. v = frp_divisor_cfba_determinant_coprime * frp_right_factor_cfba_determinant_coprime) -> frp_divisor_cfba_determinant_coprime = 1)

Complete tactic proof in conservative notation

All 32 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

32 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 assumptionsL1–5

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

  1. L1
    intro u
  2. L2
    intro U
  3. L3
    intro v
  4. L4
    intro V
  5. L5
    intro hd
02Separate the logical casesL6–6

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

  1. L6
    cases hd
03Use earlier factsL7–13

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

  1. L7
    specialize balanced_bezout_one_implies_coprime (u)
  2. L8
    specialize balanced_bezout_one_implies_coprime (v)
  3. L9
    specialize balanced_bezout_one_implies_coprime (0)
  4. L10
    specialize balanced_bezout_one_implies_coprime (U)
  5. L11
    specialize balanced_bezout_one_implies_coprime (V)
  6. L12
    specialize balanced_bezout_one_implies_coprime (0)
  7. L13
    apply balanced_bezout_one_implies_coprime
04Calculate and transport equalitiesL14–17

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

  1. L14
    trans U * v
  2. L15
    simp [mul_comm, zero_add]
  3. L16
    trans u * V + 1
  4. L17
    symm
05Use earlier factsL18–18

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

  1. L18
    exact hd_left
06Calculate and transport equalitiesL19–19

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

  1. L19
    simp [zero_add, add_succ_left]
07Use earlier factsL20–26

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

  1. L20
    specialize balanced_bezout_one_implies_coprime (u)
  2. L21
    specialize balanced_bezout_one_implies_coprime (v)
  3. L22
    specialize balanced_bezout_one_implies_coprime (V)
  4. L23
    specialize balanced_bezout_one_implies_coprime (0)
  5. L24
    specialize balanced_bezout_one_implies_coprime (0)
  6. L25
    specialize balanced_bezout_one_implies_coprime (U)
  7. L26
    apply balanced_bezout_one_implies_coprime
08Calculate and transport equalitiesL27–30

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

  1. L27
    trans u * V
  2. L28
    simp [zero_add]
  3. L29
    trans U * v + 1
  4. L30
    symm
09Use earlier factsL31–31

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

  1. L31
    exact hd_right
10Calculate and transport equalitiesL32–32

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

  1. L32
    simp [mul_comm, zero_add, add_succ_left]

Library-wide reading audit

Original defined command ledger · 32 lines
  1. 0001intro u
  2. 0002intro U
  3. 0003intro v
  4. 0004intro V
  5. 0005intro hd
  6. 0006cases hd
  7. 0007specialize balanced_bezout_one_implies_coprime (u)
  8. 0008specialize balanced_bezout_one_implies_coprime (v)
  9. 0009specialize balanced_bezout_one_implies_coprime (0)
  10. 0010specialize balanced_bezout_one_implies_coprime (U)
  11. 0011specialize balanced_bezout_one_implies_coprime (V)
  12. 0012specialize balanced_bezout_one_implies_coprime (0)
  13. 0013apply balanced_bezout_one_implies_coprime
  14. 0014trans U * v
  15. 0015simp [mul_comm, zero_add]
  16. 0016trans u * V + 1
  17. 0017symm
  18. 0018exact hd_left
  19. 0019simp [zero_add, add_succ_left]
  20. 0020specialize balanced_bezout_one_implies_coprime (u)
  21. 0021specialize balanced_bezout_one_implies_coprime (v)
  22. 0022specialize balanced_bezout_one_implies_coprime (V)
  23. 0023specialize balanced_bezout_one_implies_coprime (0)
  24. 0024specialize balanced_bezout_one_implies_coprime (0)
  25. 0025specialize balanced_bezout_one_implies_coprime (U)
  26. 0026apply balanced_bezout_one_implies_coprime
  27. 0027trans u * V
  28. 0028simp [zero_add]
  29. 0029trans U * v + 1
  30. 0030symm
  31. 0031exact hd_right
  32. 0032simp [mul_comm, zero_add, add_succ_left]