JT0004

jordan_tuple_common_divisor_transport

Alpha v35 independently verified · alpha_closed; checked-use authorized; not Stable

Divisibility of all coordinates is extensional across tuple encodings.

Exact expanded first-order arithmetic statement

forall q b c d e k. (forall jt_index_dvd_equal jt_left_dvd_equal jt_right_dvd_equal. (exists jt_gap_dvd_equalindex. jt_gap_dvd_equalindex+S (jt_index_dvd_equal)=(k)) -> (((exists fs_h_jt_dvd_equalleft. fs_h_jt_dvd_equalleft + S (jt_left_dvd_equal) = S ((S (jt_index_dvd_equal)) * c)) /\ exists fs_q_jt_dvd_equalleft. b = fs_q_jt_dvd_equalleft * S ((S (jt_index_dvd_equal)) * c) + (jt_left_dvd_equal))) -> (((exists fs_h_jt_dvd_equalright. fs_h_jt_dvd_equalright + S (jt_right_dvd_equal) = S ((S (jt_index_dvd_equal)) * e)) /\ exists fs_q_jt_dvd_equalright. d = fs_q_jt_dvd_equalright * S ((S (jt_index_dvd_equal)) * e) + (jt_right_dvd_equal))) -> jt_left_dvd_equal=jt_right_dvd_equal) -> (forall jt_index_dvdsource jt_value_dvdsource. (exists jt_gap_dvdsourceindex. jt_gap_dvdsourceindex+S (jt_index_dvdsource)=(k)) -> (((exists fs_h_jt_dvdsourceat. fs_h_jt_dvdsourceat + S (jt_value_dvdsource) = S ((S (jt_index_dvdsource)) * c)) /\ exists fs_q_jt_dvdsourceat. b = fs_q_jt_dvdsourceat * S ((S (jt_index_dvdsource)) * c) + (jt_value_dvdsource))) -> (exists jt_factor_dvdsourcedivides. (jt_value_dvdsource)=(q)*jt_factor_dvdsourcedivides)) -> (forall jt_index_dvdtarget jt_value_dvdtarget. (exists jt_gap_dvdtargetindex. jt_gap_dvdtargetindex+S (jt_index_dvdtarget)=(k)) -> (((exists fs_h_jt_dvdtargetat. fs_h_jt_dvdtargetat + S (jt_value_dvdtarget) = S ((S (jt_index_dvdtarget)) * e)) /\ exists fs_q_jt_dvdtargetat. d = fs_q_jt_dvdtargetat * S ((S (jt_index_dvdtarget)) * e) + (jt_value_dvdtarget))) -> (exists jt_factor_dvdtargetdivides. (jt_value_dvdtarget)=(q)*jt_factor_dvdtargetdivides))

Constructive proof overview

Generated structural guide

Divisibility of all coordinates is extensional across tuple encodings.

The unchanged tactic script uses 1 declared prerequisite and contains 32 exact native proof lines.

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

Proof neighborhood

Direct dependencies

beta_at_exists Alpha theorem; checked-use authorized

Direct dependents

JT0005 jordan_primitive_tuple_transport

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

32 script commands · 6 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.

01Fix variables and assumptionsL1–10

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

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

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

  1. L11
    intro hi
  2. L12
    intro hz
03Establish haL13–17

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

  1. L13
    have ha : exists a. ((exists fs_h_jt_dvdactual. fs_h_jt_dvdactual + S (a) = S ((S (i)) * c)) /\ exists fs_q_jt_dvdactual. b = fs_q_jt_dvdactual * S ((S (i)) * c) + (a))
  2. L14
    specialize beta_at_exists (b)
  3. L15
    specialize beta_at_exists (c)
  4. L16
    specialize beta_at_exists (i)
  5. L17
    apply beta_at_exists
04Separate the logical casesL18–18

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

  1. L18
    cases ha
05Establish hvalL19–28

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

  1. L19
    have hval : x=z
  2. L20
    specialize heq (i)
  3. L21
    specialize heq (x)
  4. L22
    specialize heq (z)
  5. L23
    apply heq
  6. L24
    exact hi
  7. L25
    exact ha_witness
  8. L26
    exact hz
  9. L27
    rewrite <- hval
  10. L28
    specialize hdiv (i)
06Use earlier factsL29–32

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

  1. L29
    specialize hdiv (x)
  2. L30
    apply hdiv
  3. L31
    exact hi
  4. L32
    exact ha_witness

Library-wide reading audit

Original exact command ledger · 32 lines
  1. 0001intro q
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro k
  7. 0007intro heq
  8. 0008intro hdiv
  9. 0009intro i
  10. 0010intro z
  11. 0011intro hi
  12. 0012intro hz
  13. 0013have ha : exists a. ((exists fs_h_jt_dvdactual. fs_h_jt_dvdactual + S (a) = S ((S (i)) * c)) /\ exists fs_q_jt_dvdactual. b = fs_q_jt_dvdactual * S ((S (i)) * c) + (a))
  14. 0014specialize beta_at_exists (b)
  15. 0015specialize beta_at_exists (c)
  16. 0016specialize beta_at_exists (i)
  17. 0017apply beta_at_exists
  18. 0018cases ha
  19. 0019have hval : x=z
  20. 0020specialize heq (i)
  21. 0021specialize heq (x)
  22. 0022specialize heq (z)
  23. 0023apply heq
  24. 0024exact hi
  25. 0025exact ha_witness
  26. 0026exact hz
  27. 0027rewrite <- hval
  28. 0028specialize hdiv (i)
  29. 0029specialize hdiv (x)
  30. 0030apply hdiv
  31. 0031exact hi
  32. 0032exact ha_witness