PA002B

add_left_cancel

Stable checked-use theorem · independently closed

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

Generated structural guide

A common left addend can be cancelled.

Use the direct prerequisites add_comm, add_right_cancel as previously established PA formulas.

The proof proceeds by direct introduction and elimination.

Referenced ingredients

PA000F add_comm PA0012 add_right_cancel

Proof neighborhood

Direct dependencies

PA000F add_comm PA0012 add_right_cancel

Direct dependents

PA002D positive_quotient_gap_impossible PA002E division_remainder_unique PA003T beta_range_injective PA00AG prime_bounded_square_one_cases PA00D1 mod_eq_add_cancel_left

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

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