PF002A

fermat_four_lt_add_positive

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Adding a nonzero natural strictly increases every natural, with an explicit gap witness.

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 first-order arithmetic statement

forall a b. ~(b = 0) -> exists k. k + S a = a + b

Constructive proof overview

Generated structural guide

Adding a nonzero natural strictly increases every natural, with an explicit gap witness.

The unchanged tactic script uses 3 declared prerequisites and contains 16 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

nonzero_is_succ Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized add_succ_left Stable theorem; checked-use authorized

Direct dependents

PF002B fermat_four_root_lt_norm

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

16 script commands · 10 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro hb
02Establish hsL4–6

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

  1. L4
    have hs : exists k. b = S k
  2. L5
    apply nonzero_is_succ
  3. L6
    exact hb
03Separate the logical casesL7–7

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

  1. L7
    cases hs
04Construct an explicit witnessL8–8

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

  1. L8
    exists x
05Calculate and transport equalitiesL9–10

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

  1. L9
    rewrite hs_witness
  2. L10
    trans S (x + a)
06Use earlier factsL11–11

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

  1. L11
    apply PA4
07Calculate and transport equalitiesL12–13

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

  1. L12
    trans S (a + x)
  2. L13
    congr
08Use earlier factsL14–14

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

  1. L14
    apply add_comm
09Calculate and transport equalitiesL15–15

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

  1. L15
    symm
10Use earlier factsL16–16

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

  1. L16
    apply PA4

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hb
  4. 0004have hs : exists k. b = S k
  5. 0005apply nonzero_is_succ
  6. 0006exact hb
  7. 0007cases hs
  8. 0008exists x
  9. 0009rewrite hs_witness
  10. 0010trans S (x + a)
  11. 0011apply PA4
  12. 0012trans S (a + x)
  13. 0013congr
  14. 0014apply add_comm
  15. 0015symm
  16. 0016apply PA4