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. ∀ c. Le(a,b) → Le(c + a,c + 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 c. (exists k. k + a = b) -> exists r. r + (c + a) = c + bProof neighborhood
Direct theorem prerequisites
Direct theorem dependents
BT00R1 ceil_div_six_total BT00RI floor_ceil_complement_budget BT00UW primorial_interval_factor_prefix_shift BT00VG primorial_even_interval_divides_central BT00VH primorial_odd_interval_divides_middle BT00X1 six_block_window_decomposition_above_thirty_two BT00YA prime_square_tail_of_two_three_range BT00YB division_first_two_of_two_three_range BT010A two_lt_double_lower_six BT010P prime_contribution_interval_prefix_shift BT0110 no_bertrand_middle_contribution_interval_le_primorial_interval BT011Q bertrand_covering_intervalDefinition-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 assumptionsL1–4
02Separate the logical casesL5–5
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L5
cases h
03Construct an explicit witnessL6–6
Supply the displayed value, then prove that it has the required property.
- L6
exists x
04Calculate and transport equalitiesL7–8
05Use earlier factsL9–9
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L9
apply add_assoc
06Calculate and transport equalitiesL10–11
07Use earlier factsL12–12
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L12
apply add_comm
08Calculate and transport equalitiesL13–14
09Use earlier factsL15–15
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L15
apply add_assoc
10Calculate and transport equalitiesL16–17
11Use earlier factsL18–18
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L18
exact h_witness