TS001B

bounded_natural_absolute_difference

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

An absolute difference between two coordinates bounded by s is itself bounded by s.

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 s a c x. (exists ftcn_bound_bounded_first. ftcn_bound_bounded_first + (a) = (s)) -> (exists ftcn_bound_bounded_second. ftcn_bound_bounded_second + (c) = (s)) -> (((a) = (c) + (x) \/ (c) = (a) + (x))) -> (exists ftcn_bound_bounded_result. ftcn_bound_bounded_result + (x) = (s))

Constructive proof overview

Generated structural guide

An absolute difference between two coordinates bounded by s is itself bounded by s.

The unchanged tactic script uses 1 declared prerequisite and contains 20 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

add_assoc Stable theorem; checked-use authorized

Direct dependents

TS001M prime_floor_decoded_affine_collision_represents_prime

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

20 script commands · 13 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 assumptionsL1–7

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

  1. L1
    intro s
  2. L2
    intro a
  3. L3
    intro c
  4. L4
    intro x
  5. L5
    intro ha
  6. L6
    intro hc
  7. L7
    intro hdifference
02Separate the logical casesL8–9

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

  1. L8
    cases hdifference
  2. L9
    cases ha
03Construct an explicit witnessL10–10

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

  1. L10
    exists x1 + c
04Calculate and transport equalitiesL11–11

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

  1. L11
    trans x1 + (c + x)
05Use earlier factsL12–12

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

  1. L12
    apply add_assoc
06Calculate and transport equalitiesL13–13

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

  1. L13
    rewrite <- hdifference_left
07Use earlier factsL14–14

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

  1. L14
    exact ha_witness
08Separate the logical casesL15–15

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

  1. L15
    cases hc
09Construct an explicit witnessL16–16

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

  1. L16
    exists x1 + a
10Calculate and transport equalitiesL17–17

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

  1. L17
    trans x1 + (a + x)
11Use earlier factsL18–18

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

  1. L18
    apply add_assoc
12Calculate and transport equalitiesL19–19

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

  1. L19
    rewrite <- hdifference_right
13Use earlier factsL20–20

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

  1. L20
    exact hc_witness

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro s
  2. 0002intro a
  3. 0003intro c
  4. 0004intro x
  5. 0005intro ha
  6. 0006intro hc
  7. 0007intro hdifference
  8. 0008cases hdifference
  9. 0009cases ha
  10. 0010exists x1 + c
  11. 0011trans x1 + (c + x)
  12. 0012apply add_assoc
  13. 0013rewrite <- hdifference_left
  14. 0014exact ha_witness
  15. 0015cases hc
  16. 0016exists x1 + a
  17. 0017trans x1 + (a + x)
  18. 0018apply add_assoc
  19. 0019rewrite <- hdifference_right
  20. 0020exact hc_witness