JT001A

jordan_tuple_listed_empty

An empty actual outer list contains no representative.

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. ∀ k. ∀ B. ∀ C. ∀ D. ∀ E. ¬JordanTupleListed(b,c,k,B,C,D,E,0)

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

Definition DAG

JordanTupleListed(b,c,k,B,C,D,E,j) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order statement 
forall b c k B C D E. ~(exists jt_index_listedempty jt_code_listedempty jt_scale_listedempty. ((exists jt_gap_listedemptyindex. jt_gap_listedemptyindex+S (jt_index_listedempty)=(0)) /\ (((((((exists fs_h_jt_listedemptycode. fs_h_jt_listedemptycode + S (jt_code_listedempty) = S ((S (jt_index_listedempty)) * C)) /\ exists fs_q_jt_listedemptycode. B = fs_q_jt_listedemptycode * S ((S (jt_index_listedempty)) * C) + (jt_code_listedempty))) /\ (((exists fs_h_jt_listedemptyscale. fs_h_jt_listedemptyscale + S (jt_scale_listedempty) = S ((S (jt_index_listedempty)) * E)) /\ exists fs_q_jt_listedemptyscale. D = fs_q_jt_listedemptyscale * S ((S (jt_index_listedempty)) * E) + (jt_scale_listedempty))))) /\ (forall jt_index_listedemptyequal jt_left_listedemptyequal jt_right_listedemptyequal. (exists jt_gap_listedemptyequalindex. jt_gap_listedemptyequalindex+S (jt_index_listedemptyequal)=(k)) -> (((exists fs_h_jt_listedemptyequalleft. fs_h_jt_listedemptyequalleft + S (jt_left_listedemptyequal) = S ((S (jt_index_listedemptyequal)) * c)) /\ exists fs_q_jt_listedemptyequalleft. b = fs_q_jt_listedemptyequalleft * S ((S (jt_index_listedemptyequal)) * c) + (jt_left_listedemptyequal))) -> (((exists fs_h_jt_listedemptyequalright. fs_h_jt_listedemptyequalright + S (jt_right_listedemptyequal) = S ((S (jt_index_listedemptyequal)) * jt_scale_listedempty)) /\ exists fs_q_jt_listedemptyequalright. jt_code_listedempty = fs_q_jt_listedemptyequalright * S ((S (jt_index_listedemptyequal)) * jt_scale_listedempty) + (jt_right_listedemptyequal))) -> jt_left_listedemptyequal=jt_right_listedemptyequal)))))

Complete tactic proof in conservative notation

All 19 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

19 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 assumptionsL1–8

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro k
  4. L4
    intro B
  5. L5
    intro C
  6. L6
    intro D
  7. L7
    intro E
  8. L8
    intro h
02Separate the logical casesL9–13

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

  1. L9
    cases h
  2. L10
    cases h_witness
  3. L11
    cases h_witness_witness
  4. L12
    cases h_witness_witness_witness
  5. L13
    cases h_witness_witness_witness_right
03Use earlier factsL14–19

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

  1. L14
    specialize lt_not_le (x)
  2. L15
    specialize lt_not_le (0)
  3. L16
    apply lt_not_le
  4. L17
    exact h_witness_witness_witness_left
  5. L18
    specialize zero_le (x)
  6. L19
    apply zero_le

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro k
  4. 0004intro B
  5. 0005intro C
  6. 0006intro D
  7. 0007intro E
  8. 0008intro h
  9. 0009cases h
  10. 0010cases h_witness
  11. 0011cases h_witness_witness
  12. 0012cases h_witness_witness_witness
  13. 0013cases h_witness_witness_witness_right
  14. 0014specialize lt_not_le (x)
  15. 0015specialize lt_not_le (0)
  16. 0016apply lt_not_le
  17. 0017exact h_witness_witness_witness_left
  18. 0018specialize zero_le (x)
  19. 0019apply zero_le