BT011O

bertrand_add_swap_nested

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

Swap the first two addends under a fixed trailing addend.

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 a b c. a + (b + c) = b + (a + c)

Structural proof guide

Swap the first two addends under a fixed trailing addend.

Direct prerequisites: add_assoc, add_comm. The authored body proceeds by direct introduction and elimination.

Proof neighborhood

Direct dependencies

BT0003 add_assoc BT0002 add_comm

Direct dependents

BT011P bertrand_add_six_permute BT0121 bertrand_cover_one_hundred_sixty_three_three_hundred_seventeen BT0123 bertrand_cutoff_lt_final_prime

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

11 script commands · 7 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.

Named ingredients (2)
add assocadd comm
01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
02Calculate and transport equalitiesL4–5

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

  1. L4
    trans (a + b) + c
  2. L5
    symm
03Use earlier factsL6–6

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

  1. L6
    apply add_assoc
04Calculate and transport equalitiesL7–8

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

  1. L7
    trans (b + a) + c
  2. L8
    congr
05Use earlier factsL9–9

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

  1. L9
    apply add_comm
06Calculate and transport equalitiesL10–10

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

  1. L10
    refl
07Use earlier factsL11–11

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

  1. L11
    apply add_assoc

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004trans (a + b) + c
  5. 0005symm
  6. 0006apply add_assoc
  7. 0007trans (b + a) + c
  8. 0008congr
  9. 0009apply add_comm
  10. 0010refl
  11. 0011apply add_assoc