PF001H · theorem body

pythagorean_positive_add_strict

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

Adding a positive natural gives an explicitly witnessed strict increase.

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 gap. gap + 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

PF001I pythagorean_leg_strictly_below_hypotenuse PF001T pythagorean_positive_gap_orders_parameters

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

11 script commands · 6 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
02Use earlier factsL4–4

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

  1. L4
    specialize nonzero_is_succ b
03Establish hsuccessorL5–7

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

  1. L5
    have hsuccessor : exists k. b = S k
  2. L6
    apply nonzero_is_succ
  3. L7
    exact hb
04Separate the logical casesL8–8

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

  1. L8
    cases hsuccessor
05Construct an explicit witnessL9–9

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

  1. L9
    exists x
06Calculate and transport equalitiesL10–11

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

  1. L10
    rewrite hsuccessor_witness
  2. L11
    simp [add_comm, add_succ_left]

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hb
  4. 0004specialize nonzero_is_succ b
  5. 0005have hsuccessor : exists k. b = S k
  6. 0006apply nonzero_is_succ
  7. 0007exact hb
  8. 0008cases hsuccessor
  9. 0009exists x
  10. 0010rewrite hsuccessor_witness
  11. 0011simp [add_comm, add_succ_left]