JT0016

jordan_tuple_equal_empty

All actual empty tuples are coordinatewise equal.

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. IntegerVectorZero(b,c,d,e,0)

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

Definition DAG

IntegerVectorZero(ab,ac,db,dc,l) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order statement 
forall b c d e. forall jt_index_eqempty jt_left_eqempty jt_right_eqempty. (exists jt_gap_eqemptyindex. jt_gap_eqemptyindex+S (jt_index_eqempty)=(0)) -> (((exists fs_h_jt_eqemptyleft. fs_h_jt_eqemptyleft + S (jt_left_eqempty) = S ((S (jt_index_eqempty)) * c)) /\ exists fs_q_jt_eqemptyleft. b = fs_q_jt_eqemptyleft * S ((S (jt_index_eqempty)) * c) + (jt_left_eqempty))) -> (((exists fs_h_jt_eqemptyright. fs_h_jt_eqemptyright + S (jt_right_eqempty) = S ((S (jt_index_eqempty)) * e)) /\ exists fs_q_jt_eqemptyright. d = fs_q_jt_eqemptyright * S ((S (jt_index_eqempty)) * e) + (jt_right_eqempty))) -> jt_left_eqempty=jt_right_eqempty

Complete tactic proof in conservative notation

All 17 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

17 script commands · 3 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 i
  6. L6
    intro a
  7. L7
    intro z
  8. L8
    intro hi
  9. L9
    intro ha
  10. L10
    intro hz
02Separate the logical casesL11–11

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

  1. L11
    exfalso
03Use earlier factsL12–17

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

  1. L12
    specialize lt_not_le (i)
  2. L13
    specialize lt_not_le (0)
  3. L14
    apply lt_not_le
  4. L15
    exact hi
  5. L16
    specialize zero_le (i)
  6. L17
    apply zero_le

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro i
  6. 0006intro a
  7. 0007intro z
  8. 0008intro hi
  9. 0009intro ha
  10. 0010intro hz
  11. 0011exfalso
  12. 0012specialize lt_not_le (i)
  13. 0013specialize lt_not_le (0)
  14. 0014apply lt_not_le
  15. 0015exact hi
  16. 0016specialize zero_le (i)
  17. 0017apply zero_le