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. 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
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
BT0057 beta_value_lt_scaled_base BT0058 new_value_lt_scaled_base BT005E beta_prefix_product_trace_exists BT005K beta_product_succ_append BT0089 beta_prefix_sum_trace_exists BT00QF prime_power_exponent_le BT00TC choose_functional BT00TE choose_zero BT00TI choose_succ_succ_of_lt BT00TJ choose_succ_succDefinition-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 assumptionsL1–3
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