PA003B

le_or_lt

Stable checked-use theorem · independently closed

Any two naturals satisfy weak order in one direction or strict order in the other.

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 PA statement

forall a b. (exists k. k + a = b) \/ exists k. k + S b = a

Structural proof guide

Generated structural guide

Any two naturals satisfy weak order in one direction or strict order in the other.

This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.

The proof proceeds by structural induction (2), case analysis (3), equality transport (2).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

PA0070 gauss_pointwise_signed_half_representative PA00DR odd_half_division_quotient_bounded PA00DS nonzero_remainder_division_positive_multiple_threshold PA00DT eisenstein_row_indicator_prefix_to_initial_segment

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

Read the argument

Proof checkpoints

26 script commands · 19 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.

01Induction on aL1–2

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

  1. L1
    induction a
  2. L2
    intro b
02Separate the logical casesL3–3

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

  1. L3
    left
03Construct an explicit witnessL4–4

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

  1. L4
    exists b
04Use earlier factsL5–5

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

  1. L5
    apply PA3
05Induction on bL6–6

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

  1. L6
    induction b
06Separate the logical casesL7–7

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

  1. L7
    right
07Construct an explicit witnessL8–8

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

  1. L8
    exists a
08Calculate and transport equalitiesL9–9

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

  1. L9
    trans S (a + 0)
09Use earlier factsL10–10

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

  1. L10
    apply PA4
10Calculate and transport equalitiesL11–11

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

  1. L11
    congr
11Use earlier factsL12–13

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

  1. L12
    apply PA3
  2. L13
    specialize IH b
12Separate the logical casesL14–16

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

  1. L14
    cases IH
  2. L15
    left
  3. L16
    cases IH_left
13Construct an explicit witnessL17–17

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

  1. L17
    exists x
14Calculate and transport equalitiesL18–19

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

  1. L18
    rewrite PA4
  2. L19
    congr
15Use earlier factsL20–20

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

  1. L20
    exact IH_left_witness
16Separate the logical casesL21–22

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

  1. L21
    right
  2. L22
    cases IH_right
17Construct an explicit witnessL23–23

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

  1. L23
    exists x
18Calculate and transport equalitiesL24–25

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

  1. L24
    rewrite PA4
  2. L25
    congr
19Use earlier factsL26–26

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

  1. L26
    exact IH_right_witness

Library-wide reading audit

Original exact command ledger · 26 lines
  1. 0001induction a
  2. 0002intro b
  3. 0003left
  4. 0004exists b
  5. 0005apply PA3
  6. 0006induction b
  7. 0007right
  8. 0008exists a
  9. 0009trans S (a + 0)
  10. 0010apply PA4
  11. 0011congr
  12. 0012apply PA3
  13. 0013specialize IH b
  14. 0014cases IH
  15. 0015left
  16. 0016cases IH_left
  17. 0017exists x
  18. 0018rewrite PA4
  19. 0019congr
  20. 0020exact IH_left_witness
  21. 0021right
  22. 0022cases IH_right
  23. 0023exists x
  24. 0024rewrite PA4
  25. 0025congr
  26. 0026exact IH_right_witness