BT0025 · Bertrand theorem

prime_two

Stable checked-use theorem · independently kernel verified

Two is prime in the expanded first-order prime predicate.

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

Prime(2)

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

PD0004 Prime

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
~(2 = 1) /\ forall a b. 2 = a * b -> a = 1 \/ b = 1

Proof neighborhood

Direct theorem prerequisites

BT0004 mul_zero_left BT0024 two_large_factors_impossible

Direct theorem dependents

BT0124 bertrand_small_closed_upper

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

Read the argument

Proof checkpoints

35 script commands · 18 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)
mul zero lefttwo large factors impossible
01Separate the logical casesL1–1

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

  1. L1
    split
02Fix variables and assumptionsL2–2

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

  1. L2
    intro h
03Establish h10L3–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply PA2.

  1. L3
    have h10 : 1 = 0
  2. L4
    apply PA2
  3. L5
    exact h
  4. L6
    apply PA1
  5. L7
    exact h10
  6. L8
    intro a
04Induction on bL9–11

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L9
    induction b
  2. L10
    intro h
  3. L11
    rewrite PA5 at h
05Separate the logical casesL12–12

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

  1. L12
    exfalso
06Use earlier factsL13–14

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

  1. L13
    apply PA1
  2. L14
    exact h
07Induction on bL15–16

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L15
    induction b
  2. L16
    intro h
08Separate the logical casesL17–17

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

  1. L17
    right
09Calculate and transport equalitiesL18–18

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

  1. L18
    refl
10Induction on aL19–22

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L19
    induction a
  2. L20
    intro h
  3. L21
    specialize mul_zero_left (S (S b))
  4. L22
    rewrite mul_zero_left at h
11Separate the logical casesL23–23

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

  1. L23
    exfalso
12Use earlier factsL24–25

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

  1. L24
    apply PA1
  2. L25
    exact h
13Induction on aL26–27

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L26
    induction a
  2. L27
    intro h
14Separate the logical casesL28–28

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

  1. L28
    left
15Calculate and transport equalitiesL29–29

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

  1. L29
    refl
16Fix variables and assumptionsL30–30

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

  1. L30
    intro h
17Separate the logical casesL31–31

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

  1. L31
    exfalso
18Use earlier factsL32–35

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

  1. L32
    specialize two_large_factors_impossible a
  2. L33
    specialize two_large_factors_impossible b
  3. L34
    apply two_large_factors_impossible
  4. L35
    exact h

Library-wide reading audit

Original defined command ledger · 35 lines
  1. 0001split
  2. 0002intro h
  3. 0003have h10 : 1 = 0
  4. 0004apply PA2
  5. 0005exact h
  6. 0006apply PA1
  7. 0007exact h10
  8. 0008intro a
  9. 0009induction b
  10. 0010intro h
  11. 0011rewrite PA5 at h
  12. 0012exfalso
  13. 0013apply PA1
  14. 0014exact h
  15. 0015induction b
  16. 0016intro h
  17. 0017right
  18. 0018refl
  19. 0019induction a
  20. 0020intro h
  21. 0021specialize mul_zero_left (S (S b))
  22. 0022rewrite mul_zero_left at h
  23. 0023exfalso
  24. 0024apply PA1
  25. 0025exact h
  26. 0026induction a
  27. 0027intro h
  28. 0028left
  29. 0029refl
  30. 0030intro h
  31. 0031exfalso
  32. 0032specialize two_large_factors_impossible a
  33. 0033specialize two_large_factors_impossible b
  34. 0034apply two_large_factors_impossible
  35. 0035exact h