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
∀ a. ∀ b. Le(a,b) → Lt(a,S b)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
2 occurrences
In local proof propositions
0 occurrences
Exact expanded native-PA statement
forall a b. (exists k. k + a = b) -> exists r. r + S a = S bProof neighborhood
Direct theorem prerequisites
Direct theorem dependents
PA002Q new_value_lt_scaled_base PA002T beta_value_lt_scaled_base PA003G beta_prefix_sum_trace_exists PA003W beta_prefix_product_trace_exists PA008B prime_mul_index_map_exists_up_to PA008E prime_mul_index_map_injective PA008K prime_range_product_coprime PA008R prime_scaled_inverse_prefix_extend PA00A4 prime_inverse_index_exists PA00AH prime_inverse_prefix_fixed_cases PA00AL bounded_inverse_index_uniqueDefinition-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 assumptionsL1–3
02Separate the logical casesL4–4
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L4
cases h
03Construct an explicit witnessL5–5
Supply the displayed value, then prove that it has the required property.
- L5
exists x
04Calculate and transport equalitiesL6–7
05Use earlier factsL8–8
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L8
exact h_witness