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
∀ p. ∀ a. ∀ x. (ScaledInverse(p,a,x,x) → ScaledFixedPoint(p,a,x)) ∧ (ScaledFixedPoint(p,a,x) → ScaledInverse(p,a,x,x))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
4 occurrences
In local proof propositions
0 occurrences
Exact expanded native-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))))))Proof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-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
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