MD000A

signed_matrix_two_determinant_functional

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The positive/negative natural components of a 2-by-2 determinant are unique.

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 first-order arithmetic statement

forall a b c d p n q m. (p = a * d /\ n = b * c) -> (q = a * d /\ m = b * c) -> (p = q /\ n = m)

Constructive proof overview

Generated structural guide

The positive/negative natural components of a 2-by-2 determinant are unique.

The unchanged tactic script uses 0 declared prerequisites and contains 21 exact native proof lines.

Alpha v34 checked-use · first admitted v20 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

none

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

21 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro p
  6. L6
    intro n
  7. L7
    intro q
  8. L8
    intro m
  9. L9
    intro hp
  10. L10
    intro hq
02Separate the logical casesL11–13

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

  1. L11
    cases hp
  2. L12
    cases hq
  3. L13
    split
03Calculate and transport equalitiesL14–14

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

  1. L14
    trans a * d
04Use earlier factsL15–15

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

  1. L15
    exact hp_left
05Calculate and transport equalitiesL16–16

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

  1. L16
    symm
06Use earlier factsL17–17

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

  1. L17
    exact hq_left
07Calculate and transport equalitiesL18–18

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

  1. L18
    trans b * c
08Use earlier factsL19–19

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

  1. L19
    exact hp_right
09Calculate and transport equalitiesL20–20

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

  1. L20
    symm
10Use earlier factsL21–21

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

  1. L21
    exact hq_right

Library-wide reading audit

Original exact command ledger · 21 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro p
  6. 0006intro n
  7. 0007intro q
  8. 0008intro m
  9. 0009intro hp
  10. 0010intro hq
  11. 0011cases hp
  12. 0012cases hq
  13. 0013split
  14. 0014trans a * d
  15. 0015exact hp_left
  16. 0016symm
  17. 0017exact hq_left
  18. 0018trans b * c
  19. 0019exact hp_right
  20. 0020symm
  21. 0021exact hq_right

Separate complete second-wave branches: Full T13 proof · Alpha v27.