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
∀ b. ∀ c. ∀ z. ∀ d. ∀ a. ∀ l. Range(b,c,a,l) → Range(z,d,a,l) → ∀ x. ∀ y. Lt(x,l) → BetaAt(b,c,x,y) → BetaAt(z,d,x,y)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
5 occurrences
In local proof propositions
0 occurrences
Exact expanded native-PA statement
forall b c z d a l. (forall ff_i_transport_l. (exists ff_lt_transport_l_bound. ff_lt_transport_l_bound + S ff_i_transport_l = l) -> (((exists ff_h_transport_l_decoded. ff_h_transport_l_decoded + S (a + ff_i_transport_l) = S ((S (ff_i_transport_l)) * c)) /\ exists ff_q_transport_l_decoded. b = ff_q_transport_l_decoded * S ((S (ff_i_transport_l)) * c) + (a + ff_i_transport_l)))) -> (forall ff_i_transport_r. (exists ff_lt_transport_r_bound. ff_lt_transport_r_bound + S ff_i_transport_r = l) -> (((exists ff_h_transport_r_decoded. ff_h_transport_r_decoded + S (a + ff_i_transport_r) = S ((S (ff_i_transport_r)) * d)) /\ exists ff_q_transport_r_decoded. z = ff_q_transport_r_decoded * S ((S (ff_i_transport_r)) * d) + (a + ff_i_transport_r)))) -> forall i x. (exists h. h + S i = l) -> (((exists ff_h_range_transport_x. ff_h_range_transport_x + S (x) = S ((S (i)) * c)) /\ exists ff_q_range_transport_x. b = ff_q_range_transport_x * S ((S (i)) * c) + (x))) -> (((exists ff_h_range_transport_y. ff_h_range_transport_y + S (x) = S ((S (i)) * d)) /\ exists ff_q_range_transport_y. z = ff_q_range_transport_y * S ((S (i)) * d) + (x)))Proof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-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.
Named ingredients (1)
01Fix variables and assumptionsL1–10
02Fix variables and assumptionsL11–12
03Establish hxaL13–22
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta range entry eq.
04Use earlier factsL23–23
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L23
exact hx
05Calculate and transport equalitiesL24–25
Original defined command ledger · 28 lines
- 0001
intro b - 0002
intro c - 0003
intro z - 0004
intro d - 0005
intro a - 0006
intro l - 0007
intro hleft - 0008
intro hright - 0009
intro i - 0010
intro x - 0011
intro hi - 0012
intro hx - 0013
have hxa : x = a + i - 0014
specialize beta_range_entry_eq b - 0015
specialize beta_range_entry_eq c - 0016
specialize beta_range_entry_eq a - 0017
specialize beta_range_entry_eq l - 0018
specialize beta_range_entry_eq i - 0019
specialize beta_range_entry_eq x - 0020
apply beta_range_entry_eq - 0021
exact hleft - 0022
exact hi - 0023
exact hx - 0024
rewrite hxa - 0025
rewrite hxa - 0026
specialize hright i - 0027
apply hright - 0028
exact hi