BT004C

bezout_mod_left

Stable checked-use theorem · independently kernel verified

A balanced Bezout identity selects the right coefficient modulo the left modulus.

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.

Exact expanded PA statement

forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. n * yp + m * u = (1 + n * yn) + m * v

Structural proof guide

A balanced Bezout identity selects the right coefficient modulo the left modulus.

Direct prerequisites: add_assoc, add_comm. The authored body proceeds by direct introduction and elimination.

Proof neighborhood

Direct dependencies

BT0003 add_assoc BT0002 add_comm

Direct dependents

BT004F binary_crt

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

19 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.

Named ingredients (2)
add assocadd comm
01Fix variables and assumptionsL1–7

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

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro xp
  4. L4
    intro yp
  5. L5
    intro xn
  6. L6
    intro yn
  7. L7
    intro h
02Construct an explicit witnessL8–9

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

  1. L8
    exists xp
  2. L9
    exists xn
03Calculate and transport equalitiesL10–10

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

  1. L10
    trans m * xp + n * yp
04Use earlier factsL11–11

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

  1. L11
    apply add_comm
05Calculate and transport equalitiesL12–12

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

  1. L12
    trans 1 + (m * xn + n * yn)
06Use earlier factsL13–13

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

  1. L13
    exact h
07Calculate and transport equalitiesL14–16

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

  1. L14
    trans 1 + (n * yn + m * xn)
  2. L15
    congr
  3. L16
    refl
08Use earlier factsL17–17

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

  1. L17
    apply add_comm
09Calculate and transport equalitiesL18–18

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

  1. L18
    symm
10Use earlier factsL19–19

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

  1. L19
    apply add_assoc

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro xp
  4. 0004intro yp
  5. 0005intro xn
  6. 0006intro yn
  7. 0007intro h
  8. 0008exists xp
  9. 0009exists xn
  10. 0010trans m * xp + n * yp
  11. 0011apply add_comm
  12. 0012trans 1 + (m * xn + n * yn)
  13. 0013exact h
  14. 0014trans 1 + (n * yn + m * xn)
  15. 0015congr
  16. 0016refl
  17. 0017apply add_comm
  18. 0018symm
  19. 0019apply add_assoc