PA002B · theorem

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.

Statement with defined notation

forall a b c. a + b = a + c -> b = c

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

none

0 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall a b c. a + b = a + c -> b = c

Proof neighborhood

Direct theorem prerequisites

PA000F add_comm PA0012 add_right_cancel

Direct theorem 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

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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