TS0010 · theorem body

equal_affine_remainders_balanced

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

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

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

none

In local proof propositions

none
Exact expanded first-order 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

Proof neighborhood

Direct theorem prerequisites

add_permute_outer · Stable closed

Direct theorem dependents

TS001L equal_remainder_affine_values_balanced_congruent

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

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 assumptionsL1–10

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 defined 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