BT001X

add_eq_zero_left

Stable checked-use theorem · independently kernel verified

A sum equal to zero has zero as its left 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. a + b = 0 -> a = 0

Structural proof guide

A sum equal to zero has zero as its left addend.

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

Proof neighborhood

Direct dependencies

BT0002 add_comm BT000L add_eq_zero_right

Direct dependents

BT0024 two_large_factors_impossible BT004N bounded_common_multiple_exists BT006N prime_three

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

9 script commands · 4 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 eq zero right
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 h
02Use earlier factsL4–6

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

  1. L4
    specialize add_eq_zero_right b
  2. L5
    specialize add_eq_zero_right a
  3. L6
    apply add_eq_zero_right
03Calculate and transport equalitiesL7–7

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

  1. L7
    trans a + b
04Use earlier factsL8–9

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

  1. L8
    apply add_comm
  2. L9
    exact h

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004specialize add_eq_zero_right b
  5. 0005specialize add_eq_zero_right a
  6. 0006apply add_eq_zero_right
  7. 0007trans a + b
  8. 0008apply add_comm
  9. 0009exact h