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 p n D. ~(p = n) -> (((p) = (n) + (D)) \/ ((n) = (p) + (D))) -> ~(D = 0)Constructive proof overview
Generated structural guide
A nonzero represented integer has a nonzero actual absolute difference, in either sign orientation.
The unchanged tactic script uses 0 declared prerequisites and contains 17 exact native proof lines.
Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
Direct dependents
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
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 assumptionsL1–6
02Use earlier factsL7–7
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L7
apply hpair
03Separate the logical casesL8–8
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L8
cases habsolute
04Calculate and transport equalitiesL9–9
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L9
trans n + D
05Use earlier factsL10–10
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L10
exact habsolute_left
06Calculate and transport equalitiesL11–11
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L11
rewrite hzero
07Use earlier factsL12–12
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L12
apply PA3
08Calculate and transport equalitiesL13–14
09Use earlier factsL15–15
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L15
exact habsolute_right
10Calculate and transport equalitiesL16–16
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L16
rewrite hzero
11Use earlier factsL17–17
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L17
apply PA3