PF002A · theorem body

fermat_four_lt_add_positive

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ a. ∀ b. ¬b = 0 → Lt(a,a + b)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0002 Lt

In local proof propositions

none
Exact expanded first-order statement
forall a b. ~(b = 0) -> exists k. k + S a = a + b

Proof neighborhood

Direct theorem prerequisites

nonzero_is_succ · Stable closed add_comm · Stable closed add_succ_left · Stable closed

Direct theorem dependents

PF002B fermat_four_root_lt_norm

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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