TS001J

strict_successor_coordinate_bound_is_weak_bound

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

A coordinate strictly below S s carries an explicit weak-order witness below 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 a s. (exists ftcn_strict_successor_source. ftcn_strict_successor_source + S (a) = (S s)) -> (exists ftcn_bound_successor_result. ftcn_bound_successor_result + (a) = (s))

Constructive proof overview

Generated structural guide

A coordinate strictly below S s carries an explicit weak-order witness below s.

The unchanged tactic script uses 1 declared prerequisite and contains 7 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_of_succ_le_succ 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

7 script commands · 2 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 assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro s
  3. L3
    intro hstrict
02Use earlier factsL4–7

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

  1. L4
    specialize le_of_succ_le_succ a
  2. L5
    specialize le_of_succ_le_succ s
  3. L6
    apply le_of_succ_le_succ
  4. L7
    exact hstrict

Library-wide reading audit

Original exact command ledger · 7 lines
  1. 0001intro a
  2. 0002intro s
  3. 0003intro hstrict
  4. 0004specialize le_of_succ_le_succ a
  5. 0005specialize le_of_succ_le_succ s
  6. 0006apply le_of_succ_le_succ
  7. 0007exact hstrict