EL0022

lte_strict_difference_nonzero

A witnessed strictly positive natural difference is nonzero.

Alpha v34 checked-use · first admitted v29 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

All displayed hypotheses are required. Powers and positive differences are actual existential outputs. The proof constructs second-order correction identities and iterates the prime step; no binomial expansion or LTE oracle is assumed. The 2-adic variants remain separate open targets.

Exact theorem in conservative defined notation

∀ x. ∀ y. ∀ d. Lt(y,x) → x = y + d → ¬d = 0

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

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

lt_irrefl_expanded · checked external prerequisite
Original expanded first-order statement 
forall x y d. (exists olte_gap_strict_difference. olte_gap_strict_difference + S (y) = (x)) -> x = y + d -> ~(d = 0)

Complete tactic proof in conservative notation

All 15 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

15 script commands · 2 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 assumptionsL1–6

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro d
  4. L4
    intro hgt
  5. L5
    intro hbalance
  6. L6
    intro hdzero
02Establish heqL7–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply PA3.

  1. L7
    have heq : x = y
  2. L8
    trans y + d
  3. L9
    exact hbalance
  4. L10
    rewrite hdzero
  5. L11
    apply PA3
  6. L12
    rewrite heq at hgt
  7. L13
    specialize lt_irrefl_expanded (y)
  8. L14
    apply lt_irrefl_expanded
  9. L15
    exact hgt

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro d
  4. 0004intro hgt
  5. 0005intro hbalance
  6. 0006intro hdzero
  7. 0007have heq : x = y
  8. 0008trans y + d
  9. 0009exact hbalance
  10. 0010rewrite hdzero
  11. 0011apply PA3
  12. 0012rewrite heq at hgt
  13. 0013specialize lt_irrefl_expanded (y)
  14. 0014apply lt_irrefl_expanded
  15. 0015exact hgt