PA003C

remainder_decomposition_to_mod_eq

Stable checked-use theorem · independently closed

A directed quotient/remainder equation gives balanced congruence to its 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.

Exact expanded PA statement

forall m b q x. b = q * m + x -> exists u v. b + m * u = x + m * v

Structural proof guide

Generated structural guide

A directed quotient/remainder equation gives balanced congruence to its remainder.

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

The proof proceeds by equality transport (2).

Referenced ingredients

PA000F add_comm PA000H mul_comm

Proof neighborhood

Direct dependencies

PA000F add_comm PA000H mul_comm

Direct dependents

PA005J mod_eq_decidable_from_remainders PA005R quadratic_residue_bounded_equiv PA0063 prime_bounded_nonzero_mod_inverse PA0070 gauss_pointwise_signed_half_representative PA0086 nondivisor_canonical_remainder_exists PA008B prime_mul_index_map_exists_up_to PA008Q prime_scaled_inverse_exists PA00C5 canonical_remainder_from_mod

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

16 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.

Named ingredients (2)
add commmul comm
01Fix variables and assumptionsL1–5

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

  1. L1
    intro m
  2. L2
    intro b
  3. L3
    intro q
  4. L4
    intro x
  5. L5
    intro h
02Construct an explicit witnessL6–7

Supply the displayed value, then prove that it has the required property.

  1. L6
    exists 0
  2. L7
    exists q
03Calculate and transport equalitiesL8–10

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

  1. L8
    rewrite PA5
  2. L9
    rewrite PA3
  3. L10
    trans q * m + x
04Use earlier factsL11–11

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

  1. L11
    exact h
05Calculate and transport equalitiesL12–12

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

  1. L12
    trans x + q * m
06Use earlier factsL13–13

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

  1. L13
    apply add_comm
07Calculate and transport equalitiesL14–15

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

  1. L14
    congr
  2. L15
    refl
08Use earlier factsL16–16

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

  1. L16
    apply mul_comm

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro m
  2. 0002intro b
  3. 0003intro q
  4. 0004intro x
  5. 0005intro h
  6. 0006exists 0
  7. 0007exists q
  8. 0008rewrite PA5
  9. 0009rewrite PA3
  10. 0010trans q * m + x
  11. 0011exact h
  12. 0012trans x + q * m
  13. 0013apply add_comm
  14. 0014congr
  15. 0015refl
  16. 0016apply mul_comm