FS001V

four_square_descent_add_le_add

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

Two explicitly witnessed natural weak inequalities add constructively.

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 b c d. (exists gap. gap + a = b) -> (exists gap. gap + c = d) -> exists gap. gap + (a + c) = b + d

Constructive proof overview

Generated structural guide

Two explicitly witnessed natural weak inequalities add constructively.

The unchanged tactic script uses 1 declared prerequisite and contains 14 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

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

add_shuffle_middle Alpha theorem; checked-use authorized

Direct dependents

FS001X four_square_descent_odd_half_norm_strict

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

14 script commands · 7 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 assumptionsL1–6

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro hfirst
  6. L6
    intro hsecond
02Separate the logical casesL7–8

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

  1. L7
    cases hfirst
  2. L8
    cases hsecond
03Construct an explicit witnessL9–9

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

  1. L9
    exists x + x1
04Calculate and transport equalitiesL10–10

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

  1. L10
    trans (x + a) + (x1 + c)
05Use earlier factsL11–11

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

  1. L11
    apply add_shuffle_middle
06Calculate and transport equalitiesL12–12

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

  1. L12
    congr
07Use earlier factsL13–14

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

  1. L13
    exact hfirst_witness
  2. L14
    exact hsecond_witness

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro hfirst
  6. 0006intro hsecond
  7. 0007cases hfirst
  8. 0008cases hsecond
  9. 0009exists x + x1
  10. 0010trans (x + a) + (x1 + c)
  11. 0011apply add_shuffle_middle
  12. 0012congr
  13. 0013exact hfirst_witness
  14. 0014exact hsecond_witness