JT001F

jordan_tuple_equal_entry

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

Coordinate equality transports an actual beta entry with unchanged value.

Exact expanded first-order arithmetic 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)))

Constructive proof overview

Generated structural guide

Coordinate equality transports an actual beta entry with unchanged value.

The unchanged tactic script uses 1 declared prerequisite and contains 27 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

JT0020 jordan_tuple_bounded_transport JT0026 jordan_tuple_scan_append JT0040 jordan_rectangle_crt_append

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

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.

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 : exists z. ((exists fs_h_jt_equalentry. fs_h_jt_equalentry + S (z) = S ((S (i)) * e)) /\ exists fs_q_jt_equalentry. d = fs_q_jt_equalentry * S ((S (i)) * e) + (z))
  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 exact 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 : exists z. ((exists fs_h_jt_equalentry. fs_h_jt_equalentry + S (z) = S ((S (i)) * e)) /\ exists fs_q_jt_equalentry. d = fs_q_jt_equalentry * S ((S (i)) * e) + (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