PF001B

square_eq_injective

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

Natural square roots are unique, including zero.

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. a * a = b * b -> a = b

Constructive proof overview

Generated structural guide

Natural square roots are unique, including zero.

The unchanged tactic script uses 3 declared prerequisites and contains 18 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

PF0019 square_le_reflect le_refl Stable theorem; checked-use authorized le_antisymm Stable theorem; checked-use authorized

Direct dependents

PF001S pythagorean_half_roots_even_leg PF0020 pythagorean_ordered_gap_positive PF002S fermat_four_complete_classification

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

18 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.

Named ingredients (1)
square le reflect
01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro heq
02Use earlier factsL4–9

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

  1. L4
    specialize le_antisymm a
  2. L5
    specialize le_antisymm b
  3. L6
    apply le_antisymm
  4. L7
    specialize square_le_reflect a
  5. L8
    specialize square_le_reflect b
  6. L9
    apply square_le_reflect
03Calculate and transport equalitiesL10–10

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

  1. L10
    rewrite heq
04Use earlier factsL11–15

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

  1. L11
    specialize le_refl (b * b)
  2. L12
    apply le_refl
  3. L13
    specialize square_le_reflect b
  4. L14
    specialize square_le_reflect a
  5. L15
    apply square_le_reflect
05Calculate and transport equalitiesL16–16

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

  1. L16
    rewrite heq
06Use earlier factsL17–18

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

  1. L17
    specialize le_refl (b * b)
  2. L18
    apply le_refl

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro heq
  4. 0004specialize le_antisymm a
  5. 0005specialize le_antisymm b
  6. 0006apply le_antisymm
  7. 0007specialize square_le_reflect a
  8. 0008specialize square_le_reflect b
  9. 0009apply square_le_reflect
  10. 0010rewrite heq
  11. 0011specialize le_refl (b * b)
  12. 0012apply le_refl
  13. 0013specialize square_le_reflect b
  14. 0014specialize square_le_reflect a
  15. 0015apply square_le_reflect
  16. 0016rewrite heq
  17. 0017specialize le_refl (b * b)
  18. 0018apply le_refl