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
∀ m. ∀ n. ∀ xp. ∀ yp. ∀ xn. ∀ yn. m · xp + n · yp = 1 + (m · xn + n · yn) → ModEq(m,n · yp,1 + n · yn)Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.
Definitions used by this theorem
In the theorem statement
1 occurrences
In local proof propositions
0 occurrences
Exact expanded native-PA statement
forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. n * yp + m * u = (1 + n * yn) + m * vProof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-aware tactic body
Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.
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 assumptionsL1–7
02Construct an explicit witnessL8–9
03Calculate and transport equalitiesL10–10
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L10
trans m * xp + n * yp
04Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
apply add_comm
05Calculate and transport equalitiesL12–12
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L12
trans 1 + (m * xn + n * yn)
06Use earlier factsL13–13
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L13
exact h
07Calculate and transport equalitiesL14–16
08Use earlier factsL17–17
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L17
apply add_comm
09Calculate and transport equalitiesL18–18
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L18
symm
10Use earlier factsL19–19
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L19
apply add_assoc