PA009R

pair_index_right_below_double

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

The right position of an earlier pair lies below the doubled prefix.

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.

Exact expanded PA statement

forall t m. (exists h. h + S t = m) -> exists h. h + S (S (t + t)) = m + m

Structural proof guide

Generated structural guide

The right position of an earlier pair lies below the doubled prefix.

Use the direct prerequisites add_le_add_right, add_le_add_left, le_trans, add_succ_left as previously established PA formulas.

The proof proceeds by intermediate claims (3), equality transport (1), certified simplification (1).

Referenced ingredients

PA003J add_le_add_right PA003K add_le_add_left PA000R le_trans PA000E add_succ_left

Proof neighborhood

Direct dependencies

PA003J add_le_add_right PA003K add_le_add_left PA000R le_trans PA000E add_succ_left

Direct dependents

PA009S adjacent_scaled_orbit_history_append PA009X scaled_pair_order_successor_lift_adjacent_targets PA00AY paired_inverse_witness_append PA00B7 paired_successor_lift_adjacent_units

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Alpha-v25 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.

Read the argument

Proof checkpoints

24 script commands · 5 reading checkpoints · 3 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.

Named ingredients (3)
add le add rightadd le add leftle trans
01Fix variables and assumptionsL1–3

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

  1. L1
    intro t
  2. L2
    intro m
  3. L3
    intro htm
02Establish hfirstL4–9

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

  1. L4
    have hfirst : exists h. h + (S t + S t) = m + S t
  2. L5
    specialize add_le_add_right (S t)
  3. L6
    specialize add_le_add_right m
  4. L7
    specialize add_le_add_right (S t)
  5. L8
    apply add_le_add_right
  6. L9
    exact htm
03Establish hdoubleL10–12

Establish this local claim before using it. It is not an additional assumption.

  1. L10
    have hdouble : S (S (t + t)) = S t + S t
  2. L11
    simp [add_succ_left]
  3. L12
    rewrite <- hdouble at hfirst
04Establish hsecondL13–22

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

  1. L13
    have hsecond : exists h. h + (m + S t) = m + m
  2. L14
    specialize add_le_add_left (S t)
  3. L15
    specialize add_le_add_left m
  4. L16
    specialize add_le_add_left m
  5. L17
    apply add_le_add_left
  6. L18
    exact htm
  7. L19
    specialize le_trans (S (S (t + t)))
  8. L20
    specialize le_trans (m + S t)
  9. L21
    specialize le_trans (m + m)
  10. L22
    apply le_trans
05Use earlier factsL23–24

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

  1. L23
    exact hfirst
  2. L24
    exact hsecond

Library-wide reading audit

Original exact command ledger · 24 lines
  1. 0001intro t
  2. 0002intro m
  3. 0003intro htm
  4. 0004have hfirst : exists h. h + (S t + S t) = m + S t
  5. 0005specialize add_le_add_right (S t)
  6. 0006specialize add_le_add_right m
  7. 0007specialize add_le_add_right (S t)
  8. 0008apply add_le_add_right
  9. 0009exact htm
  10. 0010have hdouble : S (S (t + t)) = S t + S t
  11. 0011simp [add_succ_left]
  12. 0012rewrite <- hdouble at hfirst
  13. 0013have hsecond : exists h. h + (m + S t) = m + m
  14. 0014specialize add_le_add_left (S t)
  15. 0015specialize add_le_add_left m
  16. 0016specialize add_le_add_left m
  17. 0017apply add_le_add_left
  18. 0018exact htm
  19. 0019specialize le_trans (S (S (t + t)))
  20. 0020specialize le_trans (m + S t)
  21. 0021specialize le_trans (m + m)
  22. 0022apply le_trans
  23. 0023exact hfirst
  24. 0024exact hsecond