PA00FI · theorem

odd_half_of_mod4_one_exact

Alpha v34 checked-use theorem · independently closed; not Stable

The half of a fixed odd number congruent to one modulo four is exactly even.

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 p h a. p = 2 * h + 1 -> p = 4 * a + 1 -> h = 2 * a

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

none

0 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall p h a. p = 2 * h + 1 -> p = 4 * a + 1 -> h = 2 * a

Proof neighborhood

Direct theorem prerequisites

PA006R four_mul_eq_double_double PA006T odd_half_unique

Direct theorem dependents

PA00FJ odd_half_even_iff_mod4_one

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

17 script commands · 3 reading checkpoints · 1 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 (2)
four mul eq double doubleodd half unique
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro a
  4. L4
    intro hp
  5. L5
    intro hfour
02Establish hcanonicalL6–15

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

  1. L6
    have hcanonical : p = 2 * (2 * a) + 1
  2. L7
    trans 4 * a + 1
  3. L8
    exact hfour
  4. L9
    congr
  5. L10
    apply four_mul_eq_double_double
  6. L11
    refl
  7. L12
    specialize odd_half_unique p
  8. L13
    specialize odd_half_unique h
  9. L14
    specialize odd_half_unique (2 * a)
  10. L15
    apply odd_half_unique
03Use earlier factsL16–17

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

  1. L16
    exact hp
  2. L17
    exact hcanonical

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro a
  4. 0004intro hp
  5. 0005intro hfour
  6. 0006have hcanonical : p = 2 * (2 * a) + 1
  7. 0007trans 4 * a + 1
  8. 0008exact hfour
  9. 0009congr
  10. 0010apply four_mul_eq_double_double
  11. 0011refl
  12. 0012specialize odd_half_unique p
  13. 0013specialize odd_half_unique h
  14. 0014specialize odd_half_unique (2 * a)
  15. 0015apply odd_half_unique
  16. 0016exact hp
  17. 0017exact hcanonical