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. ∀ b. ∀ q. ∀ x. b = q · m + x → ModEq(m,b,x)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
1 occurrences
In local proof propositions
0 occurrences
Exact expanded native-PA statement
forall m b q x. b = q * m + x -> exists u v. b + m * u = x + m * vProof neighborhood
Direct theorem prerequisites
Direct theorem dependents
PA005J mod_eq_decidable_from_remainders PA005R quadratic_residue_bounded_equiv PA0063 prime_bounded_nonzero_mod_inverse PA0070 gauss_pointwise_signed_half_representative PA0086 nondivisor_canonical_remainder_exists PA008B prime_mul_index_map_exists_up_to PA008Q prime_scaled_inverse_exists PA00C5 canonical_remainder_from_modDefinition-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
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 assumptionsL1–5
02Construct an explicit witnessL6–7
03Calculate and transport equalitiesL8–10
04Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
exact h
05Calculate and transport equalitiesL12–12
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L12
trans x + q * m
06Use earlier factsL13–13
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L13
apply add_comm
07Calculate and transport equalitiesL14–15
08Use earlier factsL16–16
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L16
apply mul_comm