JT0034

jordan_rectangle_width_nonzero

A genuine index below u*v forces the rectangle width to be positive.

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

∀ u. ∀ v. ∀ p. Lt(p,u · v) → ¬v = 0

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

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order statement 
forall u v p. (exists jt_gap_widthbound. jt_gap_widthbound+S (p)=(u*v)) -> ~(v=0)

Complete tactic proof in conservative notation

All 16 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

16 script commands · 3 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro u
  2. L2
    intro v
  3. L3
    intro p
  4. L4
    intro hp
  5. L5
    intro hv
02Establish hprodL6–15

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

  1. L6
    have hprod : u*v=0
  2. L7
    rewrite hv
  3. L8
    rewrite PA5
  4. L9
    refl
  5. L10
    rewrite hprod at hp
  6. L11
    specialize lt_not_le (p)
  7. L12
    specialize lt_not_le (0)
  8. L13
    apply lt_not_le
  9. L14
    exact hp
  10. L15
    specialize zero_le (p)
03Use earlier factsL16–16

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

  1. L16
    apply zero_le

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro u
  2. 0002intro v
  3. 0003intro p
  4. 0004intro hp
  5. 0005intro hv
  6. 0006have hprod : u*v=0
  7. 0007rewrite hv
  8. 0008rewrite PA5
  9. 0009refl
  10. 0010rewrite hprod at hp
  11. 0011specialize lt_not_le (p)
  12. 0012specialize lt_not_le (0)
  13. 0013apply lt_not_le
  14. 0014exact hp
  15. 0015specialize zero_le (p)
  16. 0016apply zero_le