BT000L

add_eq_zero_right

Stable checked-use theorem · independently kernel verified

A sum equal to zero has zero as its right 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 -> b = 0

Structural proof guide

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

Direct prerequisites: none. The authored body proceeds by structural induction (1), equality transport (1).

Proof neighborhood

Direct dependencies

none

Direct dependents

BT000M mul_eq_zero BT000Y le_zero BT001X add_eq_zero_left BT0020 mul_eq_one_components BT002G factor_difference BT005C bounded_beta_exclusive_recode_invariant BT005E beta_prefix_product_trace_exists BT0069 beta_factor_divides_product BT007U beta_repeat_empty BT0084 beta_range_empty BT0089 beta_prefix_sum_trace_exists BT00S0 prime_power_quotient_prefix_exists BT00SF eisenstein_initial_segment_prefix_exists BT00SH division_successor_quotient_by_bit BT00T3 beta_pascal_zero_row_exists BT00T5 beta_pascal_row_step_exists BT00T7 beta_pascal_table_prefix_exists BT00TF beta_pascal_table_diagonal_boundary BT00U0 four_power_central_recurrence_step BT00U8 primorial_factor_prefix_exists BT00US primorial_interval_factor_prefix_exists BT00XX double_quotient_carry_prefix_exists BT00YQ prime_contribution_prefix_exists BT010L prime_contribution_interval_prefix_exists BT011K prime_eighty_three BT011L prime_one_hundred_sixty_three BT011M prime_three_hundred_seventeen BT011N prime_five_hundred_twenty_one BT0127 bertrand_strict

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 · 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.

01Fix variables and assumptionsL1–1

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

  1. L1
    intro a
02Induction on bL2–5

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L2
    induction b
  2. L3
    intro h
  3. L4
    refl
  4. L5
    intro h
03Separate the logical casesL6–6

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L6
    exfalso
04Use earlier factsL7–7

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

  1. L7
    apply PA1
05Calculate and transport equalitiesL8–8

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

  1. L8
    rewrite PA4 at h
06Use earlier factsL9–9

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

  1. L9
    exact h

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro a
  2. 0002induction b
  3. 0003intro h
  4. 0004refl
  5. 0005intro h
  6. 0006exfalso
  7. 0007apply PA1
  8. 0008rewrite PA4 at h
  9. 0009exact h