GI0007

gaussian_signed_square_functional

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

The actual nonnegative square of a signed pair is unique by cancellative natural arithmetic.

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 n s t. (((((p) * (p))) + (((n) * (n)))) = ((s) + (((((p) * (n))) + (((n) * (p))))))) -> (((((p) * (p))) + (((n) * (n)))) = ((t) + (((((p) * (n))) + (((n) * (p))))))) -> s = t

Constructive proof overview

Generated structural guide

The actual nonnegative square of a signed pair is unique by cancellative natural arithmetic.

The unchanged tactic script uses 1 declared prerequisite and contains 14 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_right_cancel Stable theorem; checked-use authorized

Direct dependents

GI0013 gaussian_signed_norm_functional GI001D gaussian_signed_square_zero_iff

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

14 script commands · 4 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 assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro s
  4. L4
    intro t
  5. L5
    intro hfirst
  6. L6
    intro hsecond
02Use earlier factsL7–10

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

  1. L7
    specialize add_right_cancel s
  2. L8
    specialize add_right_cancel t
  3. L9
    specialize add_right_cancel ((((p) * (n))) + (((n) * (p))))
  4. L10
    apply add_right_cancel
03Calculate and transport equalitiesL11–12

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

  1. L11
    trans ((((p) * (p))) + (((n) * (n))))
  2. L12
    symm
04Use earlier factsL13–14

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

  1. L13
    exact hfirst
  2. L14
    exact hsecond

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro s
  4. 0004intro t
  5. 0005intro hfirst
  6. 0006intro hsecond
  7. 0007specialize add_right_cancel s
  8. 0008specialize add_right_cancel t
  9. 0009specialize add_right_cancel ((((p) * (n))) + (((n) * (p))))
  10. 0010apply add_right_cancel
  11. 0011trans ((((p) * (p))) + (((n) * (n))))
  12. 0012symm
  13. 0013exact hfirst
  14. 0014exact hsecond