PF001B · theorem body

square_eq_injective

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

Statement with defined notation

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

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

Proof neighborhood

Direct theorem prerequisites

PF0019 square_le_reflect le_refl · Stable closed le_antisymm · Stable closed

Direct theorem dependents

PF001S pythagorean_half_roots_even_leg PF0020 pythagorean_ordered_gap_positive PF002S fermat_four_complete_classification

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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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