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 a b r t D. (((a * t) = (b * r) + (D)) \/ ((b * r) = (a * t) + (D))) -> (((a * t + b * 0) = (b * r) + (D)) \/ ((b * r) = (a * t + b * 0) + (D)))Constructive proof overview
Generated structural guide
Every natural numerator, including zero, is an actual signed numerator with negative component zero.
The unchanged tactic script uses 0 declared prerequisites and contains 15 exact native proof lines.
Alpha v34 checked-use · first admitted v29 · 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
02Separate the logical casesL7–8
03Calculate and transport equalitiesL9–10
04Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
exact h_left
05Separate the logical casesL12–12
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L12
right
06Calculate and transport equalitiesL13–13
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L13
trans a * t + D
07Use earlier factsL14–14
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L14
exact h_right
08Calculate and transport equalitiesL15–15
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L15
simp