TS001J · theorem body

strict_successor_coordinate_bound_is_weak_bound

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

Statement with defined notation

∀ a. ∀ s. Lt(a,S s)Le(a,s)

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

In local proof propositions

none
Exact expanded first-order 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))

Proof neighborhood

Direct theorem prerequisites

le_of_succ_le_succ · Stable closed

Direct theorem dependents

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

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.

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 assumptionsL1–3

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 defined 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