PA002L — one_le_of_ne_zero — defined notation

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

∀ n. ¬n = 0 → Lt(0,n)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0002 Lt

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement

forall n. ~(n = 0) -> 1 <= n

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

PA002N le_scaled_nonzero PA0039 divisor_le_nonzero PA0070 gauss_pointwise_signed_half_representative PA007B gauss_mixed_sign_scaled_source_impossible PA00B4 finite_bounded_nonendpoint_injective_coverage PA00BB pair_order_terminal_state_magnitude_range

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

8 script commands · 7 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.

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

01Induction on nL1–2

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

L1
induction n
L2
intro h

02Separate the logical casesL3–3

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

L3
exfalso

03Use earlier factsL4–4

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

L4
apply h

04Calculate and transport equalitiesL5–5

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

L5
refl

05Fix variables and assumptionsL6–6

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

L6
intro h

06Construct an explicit witnessL7–7

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

L7
exists n

07Calculate and transport equalitiesL8–8

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

L8
simp

Library-wide reading audit

Original defined command ledger · 8 lines