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. Lt(a,S b) → Le(a,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 + S a = S b) -> exists r. r + a = bProof neighborhood
Direct theorem prerequisites
Direct theorem dependents
PA001K gcd_balanced_bezout_exists_up_to PA001U beta_exclusive_accumulated_product_step PA002G beta_exclusive_recode_congruence_step PA002Y beta_range_succ_extend PA0035 gcd_exists_up_to PA003D finite_lt_succ_eq_or_lt PA003G beta_prefix_sum_trace_exists PA003W beta_prefix_product_trace_exists PA0044 beta_repeat_succ_extend PA008B prime_mul_index_map_exists_up_to PA009A scaled_inverse_prefix_mate_predecessor PA00A4 prime_inverse_index_exists PA00B4 finite_bounded_nonendpoint_injective_coverage PA00BB pair_order_terminal_state_magnitude_range PA00DX eisenstein_initial_segment_bit_count_functionalDefinition-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
04Use earlier factsL6–6
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L6
apply PA2