PF001R · theorem body

pythagorean_half_roots_coordinates

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

The roots of the half-factors give the hypotenuse sum and the exact subtraction-free odd-leg gap.

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 c t m n. c = a + 2 * t -> a + t = m * m -> t = n * n -> (c = m * m + n * n /\ m * m = n * n + a)

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 c t m n. c = a + 2 * t -> a + t = m * m -> t = n * n -> (c = m * m + n * n /\ m * m = n * n + a)

Proof neighborhood

Direct theorem prerequisites

PF001L pythagorean_half_hypotenuse_reassociation add_comm · Stable closed

Direct theorem dependents

PF001W pythagorean_half_factors_extract_parameters

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

24 script commands · 13 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)
pythagorean half hypotenuse reassociation
01Fix variables and assumptionsL1–8

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

  1. L1
    intro a
  2. L2
    intro c
  3. L3
    intro t
  4. L4
    intro m
  5. L5
    intro n
  6. L6
    intro hc
  7. L7
    intro hm
  8. L8
    intro hn
02Separate the logical casesL9–9

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    split
03Calculate and transport equalitiesL10–11

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

  1. L10
    trans (a + t) + t
  2. L11
    trans a + 2 * t
04Use earlier factsL12–13

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

  1. L12
    exact hc
  2. L13
    apply pythagorean_half_hypotenuse_reassociation
05Calculate and transport equalitiesL14–14

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

  1. L14
    congr
06Use earlier factsL15–16

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

  1. L15
    exact hm
  2. L16
    exact hn
07Calculate and transport equalitiesL17–18

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

  1. L17
    trans a + t
  2. L18
    symm
08Use earlier factsL19–19

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

  1. L19
    exact hm
09Calculate and transport equalitiesL20–20

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

  1. L20
    trans t + a
10Use earlier factsL21–21

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

  1. L21
    apply add_comm
11Calculate and transport equalitiesL22–22

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

  1. L22
    congr
12Use earlier factsL23–23

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

  1. L23
    exact hn
13Calculate and transport equalitiesL24–24

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

  1. L24
    refl

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro a
  2. 0002intro c
  3. 0003intro t
  4. 0004intro m
  5. 0005intro n
  6. 0006intro hc
  7. 0007intro hm
  8. 0008intro hn
  9. 0009split
  10. 0010trans (a + t) + t
  11. 0011trans a + 2 * t
  12. 0012exact hc
  13. 0013apply pythagorean_half_hypotenuse_reassociation
  14. 0014congr
  15. 0015exact hm
  16. 0016exact hn
  17. 0017trans a + t
  18. 0018symm
  19. 0019exact hm
  20. 0020trans t + a
  21. 0021apply add_comm
  22. 0022congr
  23. 0023exact hn
  24. 0024refl