JT0020

jordan_tuple_bounded_transport

The canonical coordinate bound is independent of tuple encoding.

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. ∀ n. IntegerVectorZero(b,c,d,e,k) → BetaPrefixInto(b,c,k,n) → BetaPrefixInto(d,e,k,n)

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

Definition DAG

IntegerVectorZero(ab,ac,db,dc,l) · 1BetaPrefixInto(b,c,l,B) · 2Lt(a,b) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

jordan_tuple_equal_entry
Original expanded first-order statement 
forall b c d e k n. (forall jt_index_boundequal jt_left_boundequal jt_right_boundequal. (exists jt_gap_boundequalindex. jt_gap_boundequalindex+S (jt_index_boundequal)=(k)) -> (((exists fs_h_jt_boundequalleft. fs_h_jt_boundequalleft + S (jt_left_boundequal) = S ((S (jt_index_boundequal)) * c)) /\ exists fs_q_jt_boundequalleft. b = fs_q_jt_boundequalleft * S ((S (jt_index_boundequal)) * c) + (jt_left_boundequal))) -> (((exists fs_h_jt_boundequalright. fs_h_jt_boundequalright + S (jt_right_boundequal) = S ((S (jt_index_boundequal)) * e)) /\ exists fs_q_jt_boundequalright. d = fs_q_jt_boundequalright * S ((S (jt_index_boundequal)) * e) + (jt_right_boundequal))) -> jt_left_boundequal=jt_right_boundequal) -> (forall jt_index_boundsource. (exists jt_gap_boundsourceindex. jt_gap_boundsourceindex+S (jt_index_boundsource)=(k)) -> exists jt_value_boundsource. ((((exists fs_h_jt_boundsourceat. fs_h_jt_boundsourceat + S (jt_value_boundsource) = S ((S (jt_index_boundsource)) * c)) /\ exists fs_q_jt_boundsourceat. b = fs_q_jt_boundsourceat * S ((S (jt_index_boundsource)) * c) + (jt_value_boundsource))) /\ (exists jt_gap_boundsourcevalue. jt_gap_boundsourcevalue+S (jt_value_boundsource)=(n)))) -> (forall jt_index_boundtarget. (exists jt_gap_boundtargetindex. jt_gap_boundtargetindex+S (jt_index_boundtarget)=(k)) -> exists jt_value_boundtarget. ((((exists fs_h_jt_boundtargetat. fs_h_jt_boundtargetat + S (jt_value_boundtarget) = S ((S (jt_index_boundtarget)) * e)) /\ exists fs_q_jt_boundtargetat. d = fs_q_jt_boundtargetat * S ((S (jt_index_boundtarget)) * e) + (jt_value_boundtarget))) /\ (exists jt_gap_boundtargetvalue. jt_gap_boundtargetvalue+S (jt_value_boundtarget)=(n))))

Complete tactic proof in conservative notation

All 30 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

30 script commands · 7 reading checkpoints · 1 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 entry
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 n
  7. L7
    intro heq
  8. L8
    intro hb
  9. L9
    intro i
  10. L10
    intro hi
02Establish haL11–14

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

  1. L11
    have ha : ∃ a. BetaAt(b,c,i,a) ∧ Lt(a,n)Definitions: BetaAt(b,c,i,a)Lt(a,n)Original native command in the exact edition
  2. L12
    specialize hb (i)
  3. L13
    apply hb
  4. L14
    exact hi
03Separate the logical casesL15–16

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

  1. L15
    cases ha
  2. L16
    cases ha_witness
04Construct an explicit witnessL17–17

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

  1. L17
    exists x
05Separate the logical casesL18–18

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

  1. L18
    split
06Use earlier factsL19–28

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

  1. L19
    specialize jordan_tuple_equal_entry (b)
  2. L20
    specialize jordan_tuple_equal_entry (c)
  3. L21
    specialize jordan_tuple_equal_entry (d)
  4. L22
    specialize jordan_tuple_equal_entry (e)
  5. L23
    specialize jordan_tuple_equal_entry (k)
  6. L24
    specialize jordan_tuple_equal_entry (i)
  7. L25
    specialize jordan_tuple_equal_entry (x)
  8. L26
    apply jordan_tuple_equal_entry
  9. L27
    exact heq
  10. L28
    exact hi
07Use earlier factsL29–30

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

  1. L29
    exact ha_witness_left
  2. L30
    exact ha_witness_right

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro k
  6. 0006intro n
  7. 0007intro heq
  8. 0008intro hb
  9. 0009intro i
  10. 0010intro hi
  11. 0011have ha : ∃ a. BetaAt(b,c,i,a) ∧ Lt(a,n)
  12. 0012specialize hb (i)
  13. 0013apply hb
  14. 0014exact hi
  15. 0015cases ha
  16. 0016cases ha_witness
  17. 0017exists x
  18. 0018split
  19. 0019specialize jordan_tuple_equal_entry (b)
  20. 0020specialize jordan_tuple_equal_entry (c)
  21. 0021specialize jordan_tuple_equal_entry (d)
  22. 0022specialize jordan_tuple_equal_entry (e)
  23. 0023specialize jordan_tuple_equal_entry (k)
  24. 0024specialize jordan_tuple_equal_entry (i)
  25. 0025specialize jordan_tuple_equal_entry (x)
  26. 0026apply jordan_tuple_equal_entry
  27. 0027exact heq
  28. 0028exact hi
  29. 0029exact ha_witness_left
  30. 0030exact ha_witness_right