PA0055 · theorem

even_odd_exclusive_pointwise

Stable checked-use theorem · independently closed

An even and an odd decomposition of the same natural are incompatible.

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 n a b. n = 2 * a -> n = 2 * b + 1 -> false

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 n a b. n = 2 * a -> n = 2 * b + 1 -> false

Proof neighborhood

Direct theorem prerequisites

PA002E division_remainder_unique

Direct theorem dependents

PA0056 odd_not_even PA0059 even_not_odd PA00BO double_predecessor_ne_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

24 script commands · 11 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 (1)
division remainder unique
01Fix variables and assumptionsL1–5

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

  1. L1
    intro n
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro he
  5. L5
    intro ho
02Establish huL6–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply division remainder unique.

  1. L6
    have hu : a = b /\ 0 = 1
  2. L7
    specialize division_remainder_unique 2
  3. L8
    specialize division_remainder_unique n
  4. L9
    specialize division_remainder_unique a
  5. L10
    specialize division_remainder_unique 0
  6. L11
    specialize division_remainder_unique b
  7. L12
    specialize division_remainder_unique 1
  8. L13
    apply division_remainder_unique
  9. L14
    rewrite he
  10. L15
    simp
03Construct an explicit witnessL16–16

Supply the displayed value, then prove that it has the required property.

  1. L16
    exists 1
04Calculate and transport equalitiesL17–17

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

  1. L17
    norm_num
05Use earlier factsL18–18

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

  1. L18
    exact ho
06Construct an explicit witnessL19–19

Supply the displayed value, then prove that it has the required property.

  1. L19
    exists 0
07Calculate and transport equalitiesL20–20

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

  1. L20
    norm_num
08Separate the logical casesL21–21

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

  1. L21
    cases hu
09Use earlier factsL22–22

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

  1. L22
    apply PA1
10Calculate and transport equalitiesL23–23

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

  1. L23
    symm
11Use earlier factsL24–24

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

  1. L24
    exact hu_right

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro n
  2. 0002intro a
  3. 0003intro b
  4. 0004intro he
  5. 0005intro ho
  6. 0006have hu : a = b /\ 0 = 1
  7. 0007specialize division_remainder_unique 2
  8. 0008specialize division_remainder_unique n
  9. 0009specialize division_remainder_unique a
  10. 0010specialize division_remainder_unique 0
  11. 0011specialize division_remainder_unique b
  12. 0012specialize division_remainder_unique 1
  13. 0013apply division_remainder_unique
  14. 0014rewrite he
  15. 0015simp
  16. 0016exists 1
  17. 0017norm_num
  18. 0018exact ho
  19. 0019exists 0
  20. 0020norm_num
  21. 0021cases hu
  22. 0022apply PA1
  23. 0023symm
  24. 0024exact hu_right