JT0022

jordan_tuple_listed_equal_transport

List membership is a property of the represented coordinate tuple.

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. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. IntegerVectorZero(b,c,d,e,k) → JordanTupleListed(d,e,k,B,C,D,E,j) → JordanTupleListed(b,c,k,B,C,D,E,j)

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

Definition DAG

IntegerVectorZero(ab,ac,db,dc,l) · 1JordanTupleListed(b,c,k,B,C,D,E,j) · 2

Actual proof prerequisites

jordan_tuple_equal_trans
Original expanded first-order statement 
forall b c d e k B C D E j. (forall jt_index_listedtransport jt_left_listedtransport jt_right_listedtransport. (exists jt_gap_listedtransportindex. jt_gap_listedtransportindex+S (jt_index_listedtransport)=(k)) -> (((exists fs_h_jt_listedtransportleft. fs_h_jt_listedtransportleft + S (jt_left_listedtransport) = S ((S (jt_index_listedtransport)) * c)) /\ exists fs_q_jt_listedtransportleft. b = fs_q_jt_listedtransportleft * S ((S (jt_index_listedtransport)) * c) + (jt_left_listedtransport))) -> (((exists fs_h_jt_listedtransportright. fs_h_jt_listedtransportright + S (jt_right_listedtransport) = S ((S (jt_index_listedtransport)) * e)) /\ exists fs_q_jt_listedtransportright. d = fs_q_jt_listedtransportright * S ((S (jt_index_listedtransport)) * e) + (jt_right_listedtransport))) -> jt_left_listedtransport=jt_right_listedtransport) -> (exists jt_index_listedsource jt_code_listedsource jt_scale_listedsource. ((exists jt_gap_listedsourceindex. jt_gap_listedsourceindex+S (jt_index_listedsource)=(j)) /\ (((((((exists fs_h_jt_listedsourcecode. fs_h_jt_listedsourcecode + S (jt_code_listedsource) = S ((S (jt_index_listedsource)) * C)) /\ exists fs_q_jt_listedsourcecode. B = fs_q_jt_listedsourcecode * S ((S (jt_index_listedsource)) * C) + (jt_code_listedsource))) /\ (((exists fs_h_jt_listedsourcescale. fs_h_jt_listedsourcescale + S (jt_scale_listedsource) = S ((S (jt_index_listedsource)) * E)) /\ exists fs_q_jt_listedsourcescale. D = fs_q_jt_listedsourcescale * S ((S (jt_index_listedsource)) * E) + (jt_scale_listedsource))))) /\ (forall jt_index_listedsourceequal jt_left_listedsourceequal jt_right_listedsourceequal. (exists jt_gap_listedsourceequalindex. jt_gap_listedsourceequalindex+S (jt_index_listedsourceequal)=(k)) -> (((exists fs_h_jt_listedsourceequalleft. fs_h_jt_listedsourceequalleft + S (jt_left_listedsourceequal) = S ((S (jt_index_listedsourceequal)) * e)) /\ exists fs_q_jt_listedsourceequalleft. d = fs_q_jt_listedsourceequalleft * S ((S (jt_index_listedsourceequal)) * e) + (jt_left_listedsourceequal))) -> (((exists fs_h_jt_listedsourceequalright. fs_h_jt_listedsourceequalright + S (jt_right_listedsourceequal) = S ((S (jt_index_listedsourceequal)) * jt_scale_listedsource)) /\ exists fs_q_jt_listedsourceequalright. jt_code_listedsource = fs_q_jt_listedsourceequalright * S ((S (jt_index_listedsourceequal)) * jt_scale_listedsource) + (jt_right_listedsourceequal))) -> jt_left_listedsourceequal=jt_right_listedsourceequal))))) -> (exists jt_index_listedtarget jt_code_listedtarget jt_scale_listedtarget. ((exists jt_gap_listedtargetindex. jt_gap_listedtargetindex+S (jt_index_listedtarget)=(j)) /\ (((((((exists fs_h_jt_listedtargetcode. fs_h_jt_listedtargetcode + S (jt_code_listedtarget) = S ((S (jt_index_listedtarget)) * C)) /\ exists fs_q_jt_listedtargetcode. B = fs_q_jt_listedtargetcode * S ((S (jt_index_listedtarget)) * C) + (jt_code_listedtarget))) /\ (((exists fs_h_jt_listedtargetscale. fs_h_jt_listedtargetscale + S (jt_scale_listedtarget) = S ((S (jt_index_listedtarget)) * E)) /\ exists fs_q_jt_listedtargetscale. D = fs_q_jt_listedtargetscale * S ((S (jt_index_listedtarget)) * E) + (jt_scale_listedtarget))))) /\ (forall jt_index_listedtargetequal jt_left_listedtargetequal jt_right_listedtargetequal. (exists jt_gap_listedtargetequalindex. jt_gap_listedtargetequalindex+S (jt_index_listedtargetequal)=(k)) -> (((exists fs_h_jt_listedtargetequalleft. fs_h_jt_listedtargetequalleft + S (jt_left_listedtargetequal) = S ((S (jt_index_listedtargetequal)) * c)) /\ exists fs_q_jt_listedtargetequalleft. b = fs_q_jt_listedtargetequalleft * S ((S (jt_index_listedtargetequal)) * c) + (jt_left_listedtargetequal))) -> (((exists fs_h_jt_listedtargetequalright. fs_h_jt_listedtargetequalright + S (jt_right_listedtargetequal) = S ((S (jt_index_listedtargetequal)) * jt_scale_listedtarget)) /\ exists fs_q_jt_listedtargetequalright. jt_code_listedtarget = fs_q_jt_listedtargetequalright * S ((S (jt_index_listedtargetequal)) * jt_scale_listedtarget) + (jt_right_listedtargetequal))) -> jt_left_listedtargetequal=jt_right_listedtargetequal)))))

Complete tactic proof in conservative notation

All 34 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

34 script commands · 9 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.

Named ingredients (1)
jordan tuple equal trans
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 B
  7. L7
    intro C
  8. L8
    intro D
  9. L9
    intro E
  10. L10
    intro j
02Fix variables and assumptionsL11–12

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

  1. L11
    intro heq
  2. L12
    intro hl
03Separate the logical casesL13–17

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

  1. L13
    cases hl
  2. L14
    cases hl_witness
  3. L15
    cases hl_witness_witness
  4. L16
    cases hl_witness_witness_witness
  5. L17
    cases hl_witness_witness_witness_right
04Construct an explicit witnessL18–20

Supply the displayed value, then prove that it has the required property.

  1. L18
    exists x
  2. L19
    exists x1
  3. L20
    exists x2
05Separate the logical casesL21–21

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

  1. L21
    split
06Use earlier factsL22–22

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

  1. L22
    exact hl_witness_witness_witness_left
07Separate the logical casesL23–23

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

  1. L23
    split
08Use earlier factsL24–33

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

  1. L24
    exact hl_witness_witness_witness_right_left
  2. L25
    specialize jordan_tuple_equal_trans (b)
  3. L26
    specialize jordan_tuple_equal_trans (c)
  4. L27
    specialize jordan_tuple_equal_trans (d)
  5. L28
    specialize jordan_tuple_equal_trans (e)
  6. L29
    specialize jordan_tuple_equal_trans (x1)
  7. L30
    specialize jordan_tuple_equal_trans (x2)
  8. L31
    specialize jordan_tuple_equal_trans (k)
  9. L32
    apply jordan_tuple_equal_trans
  10. L33
    exact heq
09Use earlier factsL34–34

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

  1. L34
    exact hl_witness_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 34 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro k
  6. 0006intro B
  7. 0007intro C
  8. 0008intro D
  9. 0009intro E
  10. 0010intro j
  11. 0011intro heq
  12. 0012intro hl
  13. 0013cases hl
  14. 0014cases hl_witness
  15. 0015cases hl_witness_witness
  16. 0016cases hl_witness_witness_witness
  17. 0017cases hl_witness_witness_witness_right
  18. 0018exists x
  19. 0019exists x1
  20. 0020exists x2
  21. 0021split
  22. 0022exact hl_witness_witness_witness_left
  23. 0023split
  24. 0024exact hl_witness_witness_witness_right_left
  25. 0025specialize jordan_tuple_equal_trans (b)
  26. 0026specialize jordan_tuple_equal_trans (c)
  27. 0027specialize jordan_tuple_equal_trans (d)
  28. 0028specialize jordan_tuple_equal_trans (e)
  29. 0029specialize jordan_tuple_equal_trans (x1)
  30. 0030specialize jordan_tuple_equal_trans (x2)
  31. 0031specialize jordan_tuple_equal_trans (k)
  32. 0032apply jordan_tuple_equal_trans
  33. 0033exact heq
  34. 0034exact hl_witness_witness_witness_right_right