BT000V

add_left_cancel

Stable checked-use theorem · independently kernel verified

A common left addend can be cancelled.

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 = a + c -> b = c

Structural proof guide

A common left addend can be cancelled.

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

Proof neighborhood

Direct dependencies

BT0002 add_comm BT000U add_right_cancel

Direct dependents

BT001S positive_quotient_gap_impossible BT001U division_remainder_unique BT00RG double_triple_remainder_complement_budget BT00VE primorial_interval_pairwise_coprime BT00XC add_lt_cancel_left BT00Y4 central_binom_carry_bit_count

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

13 script commands · 6 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 commadd right cancel
01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro h
02Use earlier factsL5–8

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

  1. L5
    specialize add_right_cancel b
  2. L6
    specialize add_right_cancel c
  3. L7
    specialize add_right_cancel a
  4. L8
    apply add_right_cancel
03Calculate and transport equalitiesL9–9

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

  1. L9
    trans a + b
04Use earlier factsL10–10

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

  1. L10
    apply add_comm
05Calculate and transport equalitiesL11–11

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

  1. L11
    trans a + c
06Use earlier factsL12–13

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

  1. L12
    exact h
  2. L13
    apply add_comm

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro h
  5. 0005specialize add_right_cancel b
  6. 0006specialize add_right_cancel c
  7. 0007specialize add_right_cancel a
  8. 0008apply add_right_cancel
  9. 0009trans a + b
  10. 0010apply add_comm
  11. 0011trans a + c
  12. 0012exact h
  13. 0013apply add_comm