TS001A

natural_absolute_difference_exists

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

Total natural order supplies an explicit absolute coordinate difference without subtraction.

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 c. exists x. (((a) = (c) + (x) \/ (c) = (a) + (x)))

Constructive proof overview

Generated structural guide

Total natural order supplies an explicit absolute coordinate difference without subtraction.

The unchanged tactic script uses 2 declared prerequisites and contains 19 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

le_total Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

TS001D affine_collision_absolute_difference_norm_multiple TS001X two_square_representation_multiplicatively_closed TS002G two_square_product_explicit_witness

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

19 script commands · 12 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 assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro c
02Use earlier factsL3–4

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

  1. L3
    specialize le_total a
  2. L4
    specialize le_total c
03Separate the logical casesL5–6

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

  1. L5
    cases le_total
  2. L6
    cases le_total_left
04Construct an explicit witnessL7–7

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

  1. L7
    exists x
05Separate the logical casesL8–8

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

  1. L8
    right
06Calculate and transport equalitiesL9–10

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

  1. L9
    trans x + a
  2. L10
    symm
07Use earlier factsL11–12

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

  1. L11
    exact le_total_left_witness
  2. L12
    apply add_comm
08Separate the logical casesL13–13

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

  1. L13
    cases le_total_right
09Construct an explicit witnessL14–14

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

  1. L14
    exists x
10Separate the logical casesL15–15

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

  1. L15
    left
11Calculate and transport equalitiesL16–17

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

  1. L16
    trans x + c
  2. L17
    symm
12Use earlier factsL18–19

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

  1. L18
    exact le_total_right_witness
  2. L19
    apply add_comm

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro a
  2. 0002intro c
  3. 0003specialize le_total a
  4. 0004specialize le_total c
  5. 0005cases le_total
  6. 0006cases le_total_left
  7. 0007exists x
  8. 0008right
  9. 0009trans x + a
  10. 0010symm
  11. 0011exact le_total_left_witness
  12. 0012apply add_comm
  13. 0013cases le_total_right
  14. 0014exists x
  15. 0015left
  16. 0016trans x + c
  17. 0017symm
  18. 0018exact le_total_right_witness
  19. 0019apply add_comm