JT0003

jordan_tuple_equal_trans

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

A real decoded middle coordinate witnesses transitivity.

Exact expanded first-order arithmetic statement

forall b c d e f g k. (forall jt_index_transfirst jt_left_transfirst jt_right_transfirst. (exists jt_gap_transfirstindex. jt_gap_transfirstindex+S (jt_index_transfirst)=(k)) -> (((exists fs_h_jt_transfirstleft. fs_h_jt_transfirstleft + S (jt_left_transfirst) = S ((S (jt_index_transfirst)) * c)) /\ exists fs_q_jt_transfirstleft. b = fs_q_jt_transfirstleft * S ((S (jt_index_transfirst)) * c) + (jt_left_transfirst))) -> (((exists fs_h_jt_transfirstright. fs_h_jt_transfirstright + S (jt_right_transfirst) = S ((S (jt_index_transfirst)) * e)) /\ exists fs_q_jt_transfirstright. d = fs_q_jt_transfirstright * S ((S (jt_index_transfirst)) * e) + (jt_right_transfirst))) -> jt_left_transfirst=jt_right_transfirst) -> (forall jt_index_transsecond jt_left_transsecond jt_right_transsecond. (exists jt_gap_transsecondindex. jt_gap_transsecondindex+S (jt_index_transsecond)=(k)) -> (((exists fs_h_jt_transsecondleft. fs_h_jt_transsecondleft + S (jt_left_transsecond) = S ((S (jt_index_transsecond)) * e)) /\ exists fs_q_jt_transsecondleft. d = fs_q_jt_transsecondleft * S ((S (jt_index_transsecond)) * e) + (jt_left_transsecond))) -> (((exists fs_h_jt_transsecondright. fs_h_jt_transsecondright + S (jt_right_transsecond) = S ((S (jt_index_transsecond)) * g)) /\ exists fs_q_jt_transsecondright. f = fs_q_jt_transsecondright * S ((S (jt_index_transsecond)) * g) + (jt_right_transsecond))) -> jt_left_transsecond=jt_right_transsecond) -> (forall jt_index_transresult jt_left_transresult jt_right_transresult. (exists jt_gap_transresultindex. jt_gap_transresultindex+S (jt_index_transresult)=(k)) -> (((exists fs_h_jt_transresultleft. fs_h_jt_transresultleft + S (jt_left_transresult) = S ((S (jt_index_transresult)) * c)) /\ exists fs_q_jt_transresultleft. b = fs_q_jt_transresultleft * S ((S (jt_index_transresult)) * c) + (jt_left_transresult))) -> (((exists fs_h_jt_transresultright. fs_h_jt_transresultright + S (jt_right_transresult) = S ((S (jt_index_transresult)) * g)) /\ exists fs_q_jt_transresultright. f = fs_q_jt_transresultright * S ((S (jt_index_transresult)) * g) + (jt_right_transresult))) -> jt_left_transresult=jt_right_transresult)

Constructive proof overview

Generated structural guide

A real decoded middle coordinate witnesses transitivity.

The unchanged tactic script uses 1 declared prerequisite and contains 40 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

JT0022 jordan_tuple_listed_equal_transport JT0052 jordan_enumeration_index_map_bounded_injective

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

40 script commands · 7 reading checkpoints · 3 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 b
  2. L2
    intro c
  3. L3
    intro d
  4. L4
    intro e
  5. L5
    intro f
  6. L6
    intro g
  7. L7
    intro k
  8. L8
    intro h
  9. L9
    intro hnext
  10. L10
    intro i
02Fix variables and assumptionsL11–15

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

  1. L11
    intro a
  2. L12
    intro z
  3. L13
    intro hi
  4. L14
    intro ha
  5. L15
    intro hz
03Establish hmL16–20

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

  1. L16
    have hm : exists t. ((exists fs_h_jt_middle. fs_h_jt_middle + S (t) = S ((S (i)) * e)) /\ exists fs_q_jt_middle. d = fs_q_jt_middle * S ((S (i)) * e) + (t))
  2. L17
    specialize beta_at_exists (d)
  3. L18
    specialize beta_at_exists (e)
  4. L19
    specialize beta_at_exists (i)
  5. L20
    apply beta_at_exists
04Separate the logical casesL21–21

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

  1. L21
    cases hm
05Establish hleftL22–29

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

  1. L22
    have hleft : a=x
  2. L23
    specialize h (i)
  3. L24
    specialize h (a)
  4. L25
    specialize h (x)
  5. L26
    apply h
  6. L27
    exact hi
  7. L28
    exact ha
  8. L29
    exact hm_witness
06Establish hrightL30–39

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

  1. L30
    have hright : x=z
  2. L31
    specialize hnext (i)
  3. L32
    specialize hnext (x)
  4. L33
    specialize hnext (z)
  5. L34
    apply hnext
  6. L35
    exact hi
  7. L36
    exact hm_witness
  8. L37
    exact hz
  9. L38
    trans x
  10. L39
    exact hleft
07Use earlier factsL40–40

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

  1. L40
    exact hright

Library-wide reading audit

Original exact command ledger · 40 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro f
  6. 0006intro g
  7. 0007intro k
  8. 0008intro h
  9. 0009intro hnext
  10. 0010intro i
  11. 0011intro a
  12. 0012intro z
  13. 0013intro hi
  14. 0014intro ha
  15. 0015intro hz
  16. 0016have hm : exists t. ((exists fs_h_jt_middle. fs_h_jt_middle + S (t) = S ((S (i)) * e)) /\ exists fs_q_jt_middle. d = fs_q_jt_middle * S ((S (i)) * e) + (t))
  17. 0017specialize beta_at_exists (d)
  18. 0018specialize beta_at_exists (e)
  19. 0019specialize beta_at_exists (i)
  20. 0020apply beta_at_exists
  21. 0021cases hm
  22. 0022have hleft : a=x
  23. 0023specialize h (i)
  24. 0024specialize h (a)
  25. 0025specialize h (x)
  26. 0026apply h
  27. 0027exact hi
  28. 0028exact ha
  29. 0029exact hm_witness
  30. 0030have hright : x=z
  31. 0031specialize hnext (i)
  32. 0032specialize hnext (x)
  33. 0033specialize hnext (z)
  34. 0034apply hnext
  35. 0035exact hi
  36. 0036exact hm_witness
  37. 0037exact hz
  38. 0038trans x
  39. 0039exact hleft
  40. 0040exact hright