TS002D

two_square_balanced_difference_identity

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

Equal cross products and a witnessed natural difference imply the exact balanced sum-of-two-squares identity.

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 A B C D m. A * B = C * D -> C = D + m -> (A * A + B * B) + (C * C + D * D) = (A + B) * (A + B) + m * m

Constructive proof overview

Generated structural guide

Equal cross products and a witnessed natural difference imply the exact balanced sum-of-two-squares identity.

The unchanged tactic script uses 6 declared prerequisites and contains 17 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_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized mul_add Stable theorem; checked-use authorized add_mul Stable theorem; checked-use authorized TS0029 two_square_add_left_comm

Direct dependents

TS002E two_square_product_difference_forward TS002F two_square_product_difference_reverse

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

Read the argument

Proof checkpoints

17 script commands · 3 reading checkpoints · 1 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 assumptionsL1–7

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

  1. L1
    intro A
  2. L2
    intro B
  3. L3
    intro C
  4. L4
    intro D
  5. L5
    intro m
  6. L6
    intro hcross
  7. L7
    intro hdifference
02Calculate and transport equalitiesL8–8

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

  1. L8
    rewrite hdifference at hcross
03Establish hproductL9–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul comm.

  1. L9
    have hproduct : B * A = D * D + m * D
  2. L10
    trans A * B
  3. L11
    apply mul_comm
  4. L12
    trans (D + m) * D
  5. L13
    exact hcross
  6. L14
    apply add_mul
  7. L15
    rewrite hdifference
  8. L16
    rewrite hdifference
  9. L17
    simp [mul_add, add_mul, add_assoc, add_comm, two_square_add_left_comm, mul_comm, hproduct]

Library-wide reading audit

Original exact command ledger · 17 lines
  1. 0001intro A
  2. 0002intro B
  3. 0003intro C
  4. 0004intro D
  5. 0005intro m
  6. 0006intro hcross
  7. 0007intro hdifference
  8. 0008rewrite hdifference at hcross
  9. 0009have hproduct : B * A = D * D + m * D
  10. 0010trans A * B
  11. 0011apply mul_comm
  12. 0012trans (D + m) * D
  13. 0013exact hcross
  14. 0014apply add_mul
  15. 0015rewrite hdifference
  16. 0016rewrite hdifference
  17. 0017simp [mul_add, add_mul, add_assoc, add_comm, two_square_add_left_comm, mul_comm, hproduct]