TS002D · theorem body

two_square_balanced_difference_identity

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

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.

Statement with defined notation

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

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 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

Proof neighborhood

Direct theorem prerequisites

add_assoc · Stable closed add_comm · Stable closed mul_comm · Stable closed mul_add · Stable closed add_mul · Stable closed TS0029 two_square_add_left_comm

Direct theorem dependents

TS002E two_square_product_difference_forward TS002F two_square_product_difference_reverse

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

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.

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 assumptionsL1–7

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 defined 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]