JT001F

jordan_tuple_equal_entry

Coordinate equality transports an actual beta entry with unchanged value.

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. ∀ i. ∀ a. IntegerVectorZero(b,c,d,e,k) → Lt(i,k) → BetaAt(b,c,i,a) → BetaAt(d,e,i,a)

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

Definition DAG

IntegerVectorZero(ab,ac,db,dc,l) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 3

Actual proof prerequisites

beta_at_exists · checked external prerequisite
Original expanded first-order statement 
forall b c d e k i a. (forall jt_index_entryequal jt_left_entryequal jt_right_entryequal. (exists jt_gap_entryequalindex. jt_gap_entryequalindex+S (jt_index_entryequal)=(k)) -> (((exists fs_h_jt_entryequalleft. fs_h_jt_entryequalleft + S (jt_left_entryequal) = S ((S (jt_index_entryequal)) * c)) /\ exists fs_q_jt_entryequalleft. b = fs_q_jt_entryequalleft * S ((S (jt_index_entryequal)) * c) + (jt_left_entryequal))) -> (((exists fs_h_jt_entryequalright. fs_h_jt_entryequalright + S (jt_right_entryequal) = S ((S (jt_index_entryequal)) * e)) /\ exists fs_q_jt_entryequalright. d = fs_q_jt_entryequalright * S ((S (jt_index_entryequal)) * e) + (jt_right_entryequal))) -> jt_left_entryequal=jt_right_entryequal) -> (exists jt_gap_entryindex. jt_gap_entryindex+S (i)=(k)) -> (((exists fs_h_jt_entrysource. fs_h_jt_entrysource + S (a) = S ((S (i)) * c)) /\ exists fs_q_jt_entrysource. b = fs_q_jt_entrysource * S ((S (i)) * c) + (a))) -> (((exists fs_h_jt_entrytarget. fs_h_jt_entrytarget + S (a) = S ((S (i)) * e)) /\ exists fs_q_jt_entrytarget. d = fs_q_jt_entrytarget * S ((S (i)) * e) + (a)))

Complete tactic proof in conservative notation

All 27 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

27 script commands · 5 reading checkpoints · 2 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 i
  7. L7
    intro a
  8. L8
    intro h
  9. L9
    intro hi
  10. L10
    intro ha
02Establish hzL11–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at exists.

  1. L11
    have hz : ∃ z. BetaAt(d,e,i,z)Definitions: BetaAt(d,e,i,z)Original native command in the exact edition
  2. L12
    specialize beta_at_exists (d)
  3. L13
    specialize beta_at_exists (e)
  4. L14
    specialize beta_at_exists (i)
  5. L15
    apply beta_at_exists
03Separate the logical casesL16–16

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

  1. L16
    cases hz
04Establish heqL17–26

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply h.

  1. L17
    have heq : a=x
  2. L18
    specialize h (i)
  3. L19
    specialize h (a)
  4. L20
    specialize h (x)
  5. L21
    apply h
  6. L22
    exact hi
  7. L23
    exact ha
  8. L24
    exact hz_witness
  9. L25
    rewrite heq
  10. L26
    rewrite heq
05Use earlier factsL27–27

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

  1. L27
    exact hz_witness

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro k
  6. 0006intro i
  7. 0007intro a
  8. 0008intro h
  9. 0009intro hi
  10. 0010intro ha
  11. 0011have hz : ∃ z. BetaAt(d,e,i,z)
  12. 0012specialize beta_at_exists (d)
  13. 0013specialize beta_at_exists (e)
  14. 0014specialize beta_at_exists (i)
  15. 0015apply beta_at_exists
  16. 0016cases hz
  17. 0017have heq : a=x
  18. 0018specialize h (i)
  19. 0019specialize h (a)
  20. 0020specialize h (x)
  21. 0021apply h
  22. 0022exact hi
  23. 0023exact ha
  24. 0024exact hz_witness
  25. 0025rewrite heq
  26. 0026rewrite heq
  27. 0027exact hz_witness