JT001E

jordan_tuple_prefix_equal

An actual one-way beta prefix extension preserves decoded coordinates.

Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable

95 new Alpha admissions come from 96 source lemmas: tuple equality reflexivity reuses an already-admitted theorem and is not counted twice. All counts use actual finite beta-coded enumerations. G008 multiplicativity is proved; the general prime-power count and distinct-prime product formula are further goals. General prime-power fields (G091) remain open. Stable is unchanged.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ d. ∀ e. ∀ k. BetaPrefixEqual(b,c,d,e,k) → IntegerVectorZero(b,c,d,e,k)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

IntegerVectorZero(ab,ac,db,dc,l) · 1BetaPrefixEqual(b,c,d,e,l) · 1

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c d e k. (forall jt_index_prefixsource jt_value_prefixsource. (exists jt_gap_prefixsourceindex. jt_gap_prefixsourceindex+S (jt_index_prefixsource)=(k)) -> (((exists fs_h_jt_prefixsourceold. fs_h_jt_prefixsourceold + S (jt_value_prefixsource) = S ((S (jt_index_prefixsource)) * c)) /\ exists fs_q_jt_prefixsourceold. b = fs_q_jt_prefixsourceold * S ((S (jt_index_prefixsource)) * c) + (jt_value_prefixsource))) -> (((exists fs_h_jt_prefixsourcenew. fs_h_jt_prefixsourcenew + S (jt_value_prefixsource) = S ((S (jt_index_prefixsource)) * e)) /\ exists fs_q_jt_prefixsourcenew. d = fs_q_jt_prefixsourcenew * S ((S (jt_index_prefixsource)) * e) + (jt_value_prefixsource)))) -> (forall jt_index_prefixtarget jt_left_prefixtarget jt_right_prefixtarget. (exists jt_gap_prefixtargetindex. jt_gap_prefixtargetindex+S (jt_index_prefixtarget)=(k)) -> (((exists fs_h_jt_prefixtargetleft. fs_h_jt_prefixtargetleft + S (jt_left_prefixtarget) = S ((S (jt_index_prefixtarget)) * c)) /\ exists fs_q_jt_prefixtargetleft. b = fs_q_jt_prefixtargetleft * S ((S (jt_index_prefixtarget)) * c) + (jt_left_prefixtarget))) -> (((exists fs_h_jt_prefixtargetright. fs_h_jt_prefixtargetright + S (jt_right_prefixtarget) = S ((S (jt_index_prefixtarget)) * e)) /\ exists fs_q_jt_prefixtargetright. d = fs_q_jt_prefixtargetright * S ((S (jt_index_prefixtarget)) * e) + (jt_right_prefixtarget))) -> jt_left_prefixtarget=jt_right_prefixtarget)

Complete tactic proof in conservative notation

All 24 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

24 script commands · 4 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 assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro d
  4. L4
    intro e
  5. L5
    intro k
  6. L6
    intro h
  7. L7
    intro i
  8. L8
    intro a
  9. L9
    intro z
  10. L10
    intro hi
02Fix variables and assumptionsL11–12

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

  1. L11
    intro ha
  2. L12
    intro hz
03Use earlier factsL13–22

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

  1. L13
    specialize beta_at_unique (d)
  2. L14
    specialize beta_at_unique (e)
  3. L15
    specialize beta_at_unique (i)
  4. L16
    specialize beta_at_unique (a)
  5. L17
    specialize beta_at_unique (z)
  6. L18
    apply beta_at_unique
  7. L19
    specialize h (i)
  8. L20
    specialize h (a)
  9. L21
    apply h
  10. L22
    exact hi
04Use earlier factsL23–24

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

  1. L23
    exact ha
  2. L24
    exact hz

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro k
  6. 0006intro h
  7. 0007intro i
  8. 0008intro a
  9. 0009intro z
  10. 0010intro hi
  11. 0011intro ha
  12. 0012intro hz
  13. 0013specialize beta_at_unique (d)
  14. 0014specialize beta_at_unique (e)
  15. 0015specialize beta_at_unique (i)
  16. 0016specialize beta_at_unique (a)
  17. 0017specialize beta_at_unique (z)
  18. 0018apply beta_at_unique
  19. 0019specialize h (i)
  20. 0020specialize h (a)
  21. 0021apply h
  22. 0022exact hi
  23. 0023exact ha
  24. 0024exact hz