PA001D · theorem

division_remainder_exists

Stable checked-use theorem · independently closed

Every positive divisor admits a quotient and a strictly bounded remainder.

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

∀ m. ∀ n. ¬m = 0 → ∃ x. ∃ y. DivRem(n,m,x,y)

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

PD0007 DivRem

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall m n. ~(m = 0) -> exists q r. n = m * q + r /\ S r <= m

Proof neighborhood

Direct theorem prerequisites

PA0008 zero_or_succ PA001C division_remainder_succ

Direct theorem dependents

PA001K gcd_balanced_bezout_exists_up_to PA0029 beta_at_exists PA002V mod_eq_to_remainder_decomposition PA0035 gcd_exists_up_to PA005K mod_eq_decidable_nonzero PA005R quadratic_residue_bounded_equiv PA0063 prime_bounded_nonzero_mod_inverse PA0072 gauss_half_range_signed_choices PA0086 nondivisor_canonical_remainder_exists PA008B prime_mul_index_map_exists_up_to PA008Q prime_scaled_inverse_exists PA00BY beta_division_prefix_exists

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

14 script commands · 8 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)
zero or succdivision remainder succ
01Fix variables and assumptionsL1–3

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

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro hm
02Use earlier factsL4–4

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

  1. L4
    specialize zero_or_succ m
03Separate the logical casesL5–6

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

  1. L5
    cases zero_or_succ
  2. L6
    exfalso
04Use earlier factsL7–8

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

  1. L7
    apply hm
  2. L8
    exact zero_or_succ_left
05Separate the logical casesL9–9

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

  1. L9
    cases zero_or_succ_right
06Use earlier factsL10–11

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

  1. L10
    specialize division_remainder_succ x
  2. L11
    specialize division_remainder_succ n
07Calculate and transport equalitiesL12–13

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

  1. L12
    rewrite zero_or_succ_right_witness
  2. L13
    rewrite zero_or_succ_right_witness
08Use earlier factsL14–14

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

  1. L14
    exact division_remainder_succ

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro hm
  4. 0004specialize zero_or_succ m
  5. 0005cases zero_or_succ
  6. 0006exfalso
  7. 0007apply hm
  8. 0008exact zero_or_succ_left
  9. 0009cases zero_or_succ_right
  10. 0010specialize division_remainder_succ x
  11. 0011specialize division_remainder_succ n
  12. 0012rewrite zero_or_succ_right_witness
  13. 0013rewrite zero_or_succ_right_witness
  14. 0014exact division_remainder_succ