TS0010

equal_affine_remainders_balanced

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Equal affine remainders yield an exact subtraction-free balanced congruence between their two grid values.

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 first-order arithmetic statement

forall p r i j q i2 j2 q2 t. r * i + j = p * q + t -> r * i2 + j2 = p * q2 + t -> (r * i + j) + p * q2 = (r * i2 + j2) + p * q

Constructive proof overview

Generated structural guide

Equal affine remainders yield an exact subtraction-free balanced congruence between their two grid values.

The unchanged tactic script uses 1 declared prerequisite and contains 20 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

add_permute_outer Stable theorem; checked-use authorized

Direct dependents

TS001L equal_remainder_affine_values_balanced_congruent

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

20 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro r
  3. L3
    intro i
  4. L4
    intro j
  5. L5
    intro q
  6. L6
    intro i2
  7. L7
    intro j2
  8. L8
    intro q2
  9. L9
    intro t
  10. L10
    intro hfirst
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hsecond
03Calculate and transport equalitiesL12–13

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

  1. L12
    rewrite hfirst
  2. L13
    rewrite hsecond
04Use earlier factsL14–17

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

  1. L14
    specialize add_permute_outer (p * q)
  2. L15
    specialize add_permute_outer t
  3. L16
    specialize add_permute_outer (p * q2)
  4. L17
    specialize add_permute_outer 0
05Calculate and transport equalitiesL18–19

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

  1. L18
    rewrite PA3 at add_permute_outer
  2. L19
    rewrite PA3 at add_permute_outer
06Use earlier factsL20–20

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

  1. L20
    exact add_permute_outer

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro p
  2. 0002intro r
  3. 0003intro i
  4. 0004intro j
  5. 0005intro q
  6. 0006intro i2
  7. 0007intro j2
  8. 0008intro q2
  9. 0009intro t
  10. 0010intro hfirst
  11. 0011intro hsecond
  12. 0012rewrite hfirst
  13. 0013rewrite hsecond
  14. 0014specialize add_permute_outer (p * q)
  15. 0015specialize add_permute_outer t
  16. 0016specialize add_permute_outer (p * q2)
  17. 0017specialize add_permute_outer 0
  18. 0018rewrite PA3 at add_permute_outer
  19. 0019rewrite PA3 at add_permute_outer
  20. 0020exact add_permute_outer