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. ∀ p. Lt(a,p) → Lt(b,p) → Lt(a + b,p + p)Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.
Definitions used by this theorem
In the theorem statement
In local proof propositions
Exact expanded first-order statement
forall a b p. (exists ftsp_gap_sum_first. ftsp_gap_sum_first + S (a) = p) -> (exists ftsp_gap_sum_second. ftsp_gap_sum_second + S (b) = p) -> (exists ftsp_gap_sum_result. ftsp_gap_sum_result + S (a + b) = p + p)Proof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-aware tactic body
Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay 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
02Separate the logical casesL6–7
03Construct an explicit witnessL8–8
Supply the displayed value, then prove that it has the required property.
- L8
exists x + x1 + 1
04Calculate and transport equalitiesL9–17
05Use earlier factsL18–18
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L18
apply add_assoc
06Calculate and transport equalitiesL19–20
07Use earlier factsL21–21
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L21
apply add_comm
08Calculate and transport equalitiesL22–22
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L22
refl
09Use earlier factsL23–23
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L23
apply add_assoc
Original defined command ledger · 23 lines
- 0001
intro a - 0002
intro b - 0003
intro p - 0004
intro hfirst - 0005
intro hsecond - 0006
cases hfirst - 0007
cases hsecond - 0008
exists x + x1 + 1 - 0009
rewrite <- hfirst_witness - 0010
rewrite <- hsecond_witness - 0011
simp [add_assoc, add_comm, add_succ_left] - 0012
congr - 0013
congr - 0014
congr - 0015
refl - 0016
trans (x + b) + a - 0017
symm - 0018
apply add_assoc - 0019
trans (b + x) + a - 0020
congr - 0021
apply add_comm - 0022
refl - 0023
apply add_assoc