PA00DQ · theorem

odd_half_cross_product_gap

Alpha v34 checked-use theorem · independently closed; not Stable

The odd half-products differ by the explicit positive gap h+k+1.

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.

Statement with defined notation

∀ h. ∀ k. Lt((2 · k + 1) · h,(2 · h + 1) · S k)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0002 Lt

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall h k. (exists edt_lt_gap_odd_half_cross_gap. edt_lt_gap_odd_half_cross_gap + S ((2 * k + 1) * h) = (2 * h + 1) * S k)

Proof neighborhood

Direct theorem prerequisites

PA000A mul_add PA0016 add_mul PA000B mul_assoc PA000H mul_comm PA0009 add_assoc PA000F add_comm PA000M one_mul PA0002 mul_one PA000G mul_succ_left

Direct theorem dependents

PA00DR odd_half_division_quotient_bounded

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

13 script commands · 8 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 (2)
add assocadd comm
01Fix variables and assumptionsL1–2

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

  1. L1
    intro h
  2. L2
    intro k
02Construct an explicit witnessL3–3

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

  1. L3
    exists h + k
03Calculate and transport equalitiesL4–7

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L4
    simp [mul_add, add_mul, mul_assoc, mul_comm, add_assoc, add_comm, one_mul, mul_one, mul_succ_left]
  2. L5
    congr
  3. L6
    trans (h + k) + (h + (k * h + k * h))
  4. L7
    symm
04Use earlier factsL8–8

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

  1. L8
    apply add_assoc
05Calculate and transport equalitiesL9–10

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    trans (k + h) + (h + (k * h + k * h))
  2. L10
    congr
06Use earlier factsL11–11

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

  1. L11
    apply add_comm
07Calculate and transport equalitiesL12–12

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L12
    refl
08Use earlier factsL13–13

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

  1. L13
    apply add_assoc

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro h
  2. 0002intro k
  3. 0003exists h + k
  4. 0004simp [mul_add, add_mul, mul_assoc, mul_comm, add_assoc, add_comm, one_mul, mul_one, mul_succ_left]
  5. 0005congr
  6. 0006trans (h + k) + (h + (k * h + k * h))
  7. 0007symm
  8. 0008apply add_assoc
  9. 0009trans (k + h) + (h + (k * h + k * h))
  10. 0010congr
  11. 0011apply add_comm
  12. 0012refl
  13. 0013apply add_assoc