PA0031 — prime_nonzero — 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

∀ p. Prime(p) → ¬p = 0

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

PD0004 Prime

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement

forall p. (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) -> ~(p = 0)

Proof neighborhood

Direct theorem prerequisites

PA000D mul_zero_left PA0005 succ_ne_zero

Direct theorem dependents

PA003S prime_mod_cancel PA0061 prime_is_succ_succ PA0062 prime_mod_inverse PA0063 prime_bounded_nonzero_mod_inverse PA0072 gauss_half_range_signed_choices PA007B gauss_mixed_sign_scaled_source_impossible PA0084 gauss_signed_products_cancel_mod PA008B prime_mul_index_map_exists_up_to PA008L fermat_predecessor_exponent_mod_one PA008M quadratic_residue_half_power_mod_one PA008Q prime_scaled_inverse_exists PA00BN bounded_euler_criterion_dichotomy PA00BR arbitrary_euler_criterion_residue_iff PA00BT arbitrary_euler_criterion_nonresidue_iff PA00C0 prime_scaled_half_division_prefix_exists

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

20 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.

Named ingredients (2)

mul zero left succ ne zero

01Fix variables and assumptionsL1–3

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

L1
intro p
L2
intro hp
L3
intro hp0

02Separate the logical casesL4–4

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

L4
cases hp

03Use earlier factsL5–6

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

L5
specialize hp_right 0
L6
specialize hp_right 0

04Establish hunitL7–11

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

L7
have hunit : 0 = 1 \/ 0 = 1
L8
apply hp_right
L9
rewrite hp0
L10
symm
L11
apply mul_zero_left

05Separate the logical casesL12–12

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

L12
cases hunit

06Use earlier factsL13–14

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

L13
specialize succ_ne_zero 0
L14
apply succ_ne_zero

07Calculate and transport equalitiesL15–15

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

L15
symm

08Use earlier factsL16–18

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

L16
exact hunit_left
L17
specialize succ_ne_zero 0
L18
apply succ_ne_zero

09Calculate and transport equalitiesL19–19

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

L19
symm

10Use earlier factsL20–20

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

L20
exact hunit_right

Library-wide reading audit

Original defined command ledger · 20 lines