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.
Exact expanded PA statement
forall p a x. ((((((~(x = 0) /\ (exists esi_strict_gap_fixed_relation_left_bound. esi_strict_gap_fixed_relation_left_bound + S x = p))) /\ (((~(x = 0) /\ (exists esi_strict_gap_fixed_relation_right_bound. esi_strict_gap_fixed_relation_right_bound + S x = p))) /\ (exists esi_mod_left_fixed_relation_mod esi_mod_right_fixed_relation_mod. (x * x) + p * esi_mod_left_fixed_relation_mod = (a) + p * esi_mod_right_fixed_relation_mod)))) -> ((((~(x = 0) /\ (exists esi_strict_gap_fixed_square_unit_bound. esi_strict_gap_fixed_square_unit_bound + S x = p))) /\ (exists esi_mod_left_fixed_square_square esi_mod_right_fixed_square_square. (x * x) + p * esi_mod_left_fixed_square_square = (a) + p * esi_mod_right_fixed_square_square)))) /\ (((((~(x = 0) /\ (exists esi_strict_gap_fixed_square_unit_bound. esi_strict_gap_fixed_square_unit_bound + S x = p))) /\ (exists esi_mod_left_fixed_square_square esi_mod_right_fixed_square_square. (x * x) + p * esi_mod_left_fixed_square_square = (a) + p * esi_mod_right_fixed_square_square))) -> ((((~(x = 0) /\ (exists esi_strict_gap_fixed_relation_left_bound. esi_strict_gap_fixed_relation_left_bound + S x = p))) /\ (((~(x = 0) /\ (exists esi_strict_gap_fixed_relation_right_bound. esi_strict_gap_fixed_relation_right_bound + S x = p))) /\ (exists esi_mod_left_fixed_relation_mod esi_mod_right_fixed_relation_mod. (x * x) + p * esi_mod_left_fixed_relation_mod = (a) + p * esi_mod_right_fixed_relation_mod))))))Structural proof guide
Generated structural guide
On the bounded unit domain, fixed points are exactly square roots of a.
This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.
The proof proceeds by case analysis (3).
Referenced ingredients
Proof neighborhood
Direct dependencies
Direct dependents
Formal native tactic body
Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Alpha-v25 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.
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
split
03Fix variables and assumptionsL5–5
Work with arbitrary variables or the premises of the current implication.
- L5
intro hrel
04Separate the logical casesL6–8
05Use earlier factsL9–10
06Fix variables and assumptionsL11–11
Work with arbitrary variables or the premises of the current implication.
- L11
intro hfixed
07Separate the logical casesL12–13
08Use earlier factsL14–14
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L14
exact hfixed_left
09Separate the logical casesL15–15
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L15
split