PA002A

le_total

Stable checked-use theorem · independently closed

Every pair of natural numbers is comparable in the defined order.

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 n m. n <= m \/ m <= n

Structural proof guide

Generated structural guide

Every pair of natural numbers is comparable in the defined order.

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), certified simplification (2).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

PA002E division_remainder_unique

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

23 script commands · 17 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 nL1–2

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

  1. L1
    induction n
  2. L2
    intro m
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 m
04Calculate and transport equalitiesL5–5

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

  1. L5
    simp
05Induction on mL6–6

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

  1. L6
    induction m
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 (S n)
08Calculate and transport equalitiesL9–9

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

  1. L9
    simp
09Use earlier factsL10–10

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

  1. L10
    specialize IH m
10Separate the logical casesL11–13

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

  1. L11
    cases IH
  2. L12
    cases IH_left
  3. L13
    left
11Construct an explicit witnessL14–14

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

  1. L14
    exists x
12Calculate and transport equalitiesL15–16

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

  1. L15
    rewrite PA4
  2. L16
    congr
13Use earlier factsL17–17

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

  1. L17
    exact IH_left_witness
14Separate the logical casesL18–19

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

  1. L18
    cases IH_right
  2. L19
    right
15Construct an explicit witnessL20–20

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

  1. L20
    exists x
16Calculate and transport equalitiesL21–22

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

  1. L21
    rewrite PA4
  2. L22
    congr
17Use earlier factsL23–23

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

  1. L23
    exact IH_right_witness

Library-wide reading audit

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