TS001A · theorem body

natural_absolute_difference_exists

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

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

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

none

In local proof propositions

none
Exact expanded first-order statement
forall a c. exists x. (((a) = (c) + (x) \/ (c) = (a) + (x)))

Proof neighborhood

Direct theorem prerequisites

le_total · Stable closed add_comm · Stable closed

Direct theorem dependents

TS001D affine_collision_absolute_difference_norm_multiple TS001X two_square_representation_multiplicatively_closed TS002G two_square_product_explicit_witness

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

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 assumptionsL1–2

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 defined 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