PF001I

pythagorean_leg_strictly_below_hypotenuse

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

Each leg of a Pythagorean triangle is strictly below the hypotenuse when the other leg is positive.

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. a * a + b * b = c * c -> ~(b = 0) -> exists gap. gap + S a = c

Constructive proof overview

Generated structural guide

Each leg of a Pythagorean triangle is strictly below the hypotenuse when the other leg is positive.

The unchanged tactic script uses 3 declared prerequisites and contains 19 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

PF001H pythagorean_positive_add_strict mul_ne_zero Stable theorem; checked-use authorized PF001A square_lt_reflect

Direct dependents

PF001V pythagorean_odd_even_half_factors

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

19 script commands · 3 reading checkpoints · 2 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 (2)
pythagorean positive add strictsquare lt reflect
01Fix variables and assumptionsL1–5

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 hequation
  5. L5
    intro hb
02Establish hsquareL6–11

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

  1. L6
    have hsquare : ~(b * b = 0)
  2. L7
    intro hzero
  3. L8
    apply mul_ne_zero
  4. L9
    exact hb
  5. L10
    exact hb
  6. L11
    exact hzero
03Establish hstrictL12–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean positive add strict.

  1. L12
    have hstrict : exists gap. gap + S (a * a) = c * c
  2. L13
    rewrite <- hequation
  3. L14
    specialize pythagorean_positive_add_strict (a * a)
  4. L15
    specialize pythagorean_positive_add_strict (b * b)
  5. L16
    apply pythagorean_positive_add_strict
  6. L17
    exact hsquare
  7. L18
    apply square_lt_reflect
  8. L19
    exact hstrict

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro hequation
  5. 0005intro hb
  6. 0006have hsquare : ~(b * b = 0)
  7. 0007intro hzero
  8. 0008apply mul_ne_zero
  9. 0009exact hb
  10. 0010exact hb
  11. 0011exact hzero
  12. 0012have hstrict : exists gap. gap + S (a * a) = c * c
  13. 0013rewrite <- hequation
  14. 0014specialize pythagorean_positive_add_strict (a * a)
  15. 0015specialize pythagorean_positive_add_strict (b * b)
  16. 0016apply pythagorean_positive_add_strict
  17. 0017exact hsquare
  18. 0018apply square_lt_reflect
  19. 0019exact hstrict