BT000U · Bertrand theorem

add_right_cancel

Stable checked-use theorem · independently kernel verified

A common right 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 + c = b + c -> a = b

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

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 + c = b + c -> a = b

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

BT000V add_left_cancel BT0022 mul_left_cancel_nonzero BT002G factor_difference

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

Read the argument

Proof checkpoints

13 script commands · 3 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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Induction on cL3–12

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

  1. L3
    induction c
  2. L4
    intro h
  3. L5
    rewrite PA3 at h
  4. L6
    rewrite PA3 at h
  5. L7
    exact h
  6. L8
    intro h
  7. L9
    apply IH
  8. L10
    apply PA2
  9. L11
    rewrite PA4 at h
  10. L12
    rewrite PA4 at h
03Use earlier factsL13–13

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

  1. L13
    exact h

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003induction c
  4. 0004intro h
  5. 0005rewrite PA3 at h
  6. 0006rewrite PA3 at h
  7. 0007exact h
  8. 0008intro h
  9. 0009apply IH
  10. 0010apply PA2
  11. 0011rewrite PA4 at h
  12. 0012rewrite PA4 at h
  13. 0013exact h